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## Geometry and Measurement

### Plane Geometry - Geometric Proofs

#### Q.01

'Since both sides are positive, squaring both sides gives (mb+1)² = m²+1. Therefore, m{(b²-1)m+2b}=0. Since m≠0, m=2b/(1-b²). Hence, the equation of line QR is y=2b/(1-b²)(x-b).'

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'[1] When b ≠ 1, the equation of the line QR is, with slope m, y=m(x-b), which is equivalent to mx-y-mb=0'

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'The procedure to find a trajectory [1] Represent the coordinates of any point on the trajectory as (x, y), and express the given conditions in terms of x, y. [2] Derive the equation of the trajectory and determine the geometric shape represented by that equation. [3] Verify that any point on the shape satisfies the conditions. Exclude any points on the shape that do not satisfy the conditions.'

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'[1] When ∠A=90°, the condition to be satisfied is that the y-coordinate of points A and C are equal, so t²=t²-t-1, therefore t=-1. [2] When ∠B=90°, the condition to be satisfied is that the y-coordinate of points B and C are equal, so t-2=t²-t-1, which means (t-1)²=0, thus t=1. [3] When ∠C=90°, the condition to be satisfied is AB²=BC²+CA², where AB²={(t²-(t-2))}²={(t²-t+2)}², BC²={(t+√3)-t}²+{(t²-t-1)-(t-2)}²={(t²-2t+1)}²+3, CA²={(t+√3)-t}²+{(t²-t-1)-t²}²=(t+1)²+3, thus (t²-t+2)²=(t²-2t+1)²+3+(t+1)²+3. Expanding and simplifying, we get t³-t²-t-2=0, which means (t-2)(t²+t+1)=0, and since t is a real number, t=2. Therefore, from [1] to [3], the values of t that we seek are t=-1,1,2.'

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'(2) Let point P(x, y), since AP=BP implies AP^2=BP^2,\n{% raw %}\\((x-9)^{2}+(y-10)^{2}={(x-(-5))}^{2}+(y-8)^{2}{% endraw %}\nAlso, since AP=CP implies AP^2=CP^2,\n{% raw %}\\((x-9)^{2}+(y-10)^{2}={(x-(-7))}^{2}+(y-2)^{2}{% endraw %}\nSolving this, we get P(3, 2)'

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'Consider an alternative solution, that is, when point B is fixed, the triangle ABD is fixed, so we just need to consider the case where the area of triangle BCD is maximum.'

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'The required tangent passes through point A and is not perpendicular to the x-axis, so it can be expressed as y=m(x-6)+8, which is equivalent to mx-y-6m+8=0.'

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'Important example 179 Area equality and function determination\nFind the value of the constant m for which the areas of two figures enclosed by the curve y=x^{3}-6 x^{2}+9 x and the line y=m x are equal. Here, 0<m<9.'

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'Generally, the curve $f(x, y)=0$ divides the coordinate plane into several regions (blocks). When $f(x, y)$ is a polynomial in $x, y$, the sign of $f(x, y)$ remains constant within the divided blocks.'

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'Let m be a real number. Let A, B be the intersection points of the parabola y=x^{2} and the line y=mx+1 on the coordinate plane, and O be the origin.'

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'Find the locus of point P where the sum of squares of distances from fixed points A and B is a constant value k. Assume k is greater than 0.'

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'Let the line passing through point P be denoted by ℓ, which intersects curve C at three distinct points, with x-coordinates of the intersection points being α, β, γ(α<β<γ). Prove that when the areas of the two regions enclosed by line ℓ and curve C are equal, the line ℓ passes through the origin.'

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'Find the locus of points P satisfying the condition where the difference between the squares of the distances from fixed points A and B is a constant value k. Assume k > 0.'

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'266 Practice math problem 182 => this book p. 333\n(1) The equation of line AP is\n\ny = -3px + 3p\n\nThe x-coordinate of the intersection of the line and the parabola is given by -3px + 3p = -3x^2 + 3.\nSolving it, we get x^2 - px + p - 1 = 0\nHence, (x - 1)(x - (p - 1)) = 0\nTherefore, x = 1, p - 1\nThe condition for the line segment AP and C to share a point Q different from A is that the x-coordinate of Q is p - 1.\n\n*Therefore*\n0 ≤ p - 1 < 1\n1 ≤ p < 2'

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'Find the length of the chord cut by the line y = -x + 6 on the circle x^2 + y^2 = 25 for the given chord 53 yen.'

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'Let t be a positive real number. On the xy-plane, there are two points P(t, t^{2}) and Q(-t, t^{2}+1), and a parabola C: y=x^{2}. Let f(t) be the area of the region enclosed by the line PQ and the curve C. Find the minimum value of f(t) and the corresponding value of t.'

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'The trajectory of points that satisfy the given conditions when moving forms a shape, which is called the trajectory of points that satisfy the given conditions. To demonstrate that the trajectory of points P satisfying the given conditions is shape F, two things need to be proven: 1. Any point P that satisfies the given conditions is on shape F. 2. Any point P on shape F satisfies the given conditions.'

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"Also, when point B(-2,2) is translated parallel to point O(0,0), triangle ABC moves where point A(2,3) goes to A'(4,1) and point C(1,-1) goes to C'(3,-3). Thus, triangle ABC = triangle A'OC' = 1/2 |4*(-3)-1*3| = 15/2. Next, the circumcenter of triangle ABC is the intersection of the perpendicular bisectors of edges BC and CA."

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"There is a circle O with center O and radius r. For a point P different from O, a point P' on the half-line OP with O as an endpoint is determined by OP*OP'=r^2, then associating point P with point P' on circle O is called inversion with respect to circle O, with O being the center of inversion. Furthermore, as point P moves along figure F, the figure F' traced by point P' is called the inverse of F. Regarding the inverse of circles and lines, the following properties hold: (1) The inverse of a circle passing through the center of inversion O is a line not passing through O. (2) The inverse of a line not passing through the center of inversion O is a circle passing through O. (3) The inverse of a circle not passing through the center of inversion O is a circle not passing through O. (4) The inverse of a line passing through the center of inversion O is the line itself. Example 71 on the previous page is an example of inversion with respect to the circle x^2+y^2=8, where as point P moves, the circle passed by the center of inversion O, the figure traced by point Q is the line 2x+y-4=0 not passing through O. The validity of (1)-(4) can be proved as follows."

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'On the parabola y=x^2 moving on the xy plane, two points A and B and the origin O are connected by a line segment to form triangle AOB, where ∠AOB=90°. Find the locus of the centroid G of triangle AOB.'

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"Through parallel translation that moves point P to P'(4,-3), let the point where point Q moves be Q'(x', y'). Also, let OP' = r, and let the angle between OP' and the positive direction of the x-axis be α, then r cos α = 4, r sin α = -3. Using these conditions, find the coordinates of Q'."

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'Example 51 | Maximum and minimum area of a triangle For 0 < a < sqrt{3}, there are three lines: l: y = 1-x, m: y = sqrt{3}x + 1, n: y = ax. Let A be the intersection of l and m, B be the intersection of m and n, and C be the intersection of n and l. Find the value of a that minimizes the area S of triangle ABC. Also, find the value of S at that time.'

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'The line AC is perpendicular to l, so (q-1)/(p-7) * 1/2 = -1'

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'Practice (64 => Book p.138) (1) Let a > 0, and define the coordinate axis so that A(-a, 0), B(a, 0). Let the coordinates of point P be (x, y), the given condition is AP^2 + BP^2 = k, therefore {(x+a)^2+y^2} + {(x-a)^2+y^2} = k, 4x=1 is the common internal tangent, Chapter 3 □ Practice Geometry Equations'

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'(2) When triangle PAB is formed, point P is not on the line AB. The equation of the line AB is y=-x+2. By eliminating y from this equation and y=x^2, we solve to get x=1,-2. Therefore, to form triangle PAB, it is required that s≠1, s≠-2. Let the coordinates of R be (x, y). As R is the centroid of triangle PAB, we have x=\\frac{s+3+0}{3} and y=\\frac{t-1+2}{3}, which leads to s=3x-3, t=3y-1. Substituting into (1) we get 3y-1=(3x-3)^2, i.e., y=3(x-1)^2+\\frac{1}{3}. From (3) and (4) we have x≠\\frac{4}{3}, x≠\\frac{1}{3}. Therefore, the desired trajectory is the parabola y=3(x-1)^2+\\frac{1}{3}, except for the points (\\frac{4}{3}, \\frac{2}{3}) and (\\frac{1}{3}, \\frac{5}{3}).'

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'When two tangents can be drawn from point P to the parabola y = \x0crac{1}{2} x^2$, let the two points of contact be A and B, and let the area enclosed by the segments PA, PB, and the parabola be denoted as S. Find the minimum value of S when PA and PB are perpendicular to each other.'

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'Important Example 182 Maximum and Minimum Area (3)\nWhen the curve y = | x ^ 2-x | and the line y = mx have three different intersection points, find the value of m at which the sum of the areas of the two parts enclosed by this curve and line, S, is minimized.\n[Similar question Yamagata University] <Example 169'

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'Range of passing points and curves \ a, b \ as real numbers. The parabola on the coordinate plane is the parabola \ C: y=x^{2}+a x+b \ having the parabola \ y=-x^{2} \ and two shared points, one shared point with \ x \ coordinate satisfying \ -1<x<0 \, and the other shared point with \ x \ coordinate satisfying \ 0<x<1 \. (1) Plot the possible range of point \\( (a, b) \\) on the coordinate plane. (2) Plot the possible range of parabola \ C \ on the coordinate plane. [Univ. of Tokyo]'

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'Prove the following equation: In triangle ABC, let M be the midpoint of side BC, then AB² + AC² = 2(AM² + BM²) (Median Theorem).'

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'Let a be a positive constant. Prove that the area enclosed by the tangent line at any point P on the parabola y=x^{2}+a and the parabola y=x^{2} is constant regardless of the position of point P, and find the value of that constant.'

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'Find the coordinates of the point Q that is symmetric to point P(3,7) with respect to point A(-2,-3).'

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'Find the locus of the point P such that the ratio of distances from points A(-4,0) and B(2,0) to point P is 2:1.'

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'Translate the given text into multiple languages.'

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'Find the equation of the line that bisects perpendicularly the line segment connecting points A(0,6) and B(4,4).'

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'Point P lies on the line \ y=-\\frac{1}{2 \\sqrt{2}} x+\\frac{5}{2} \, so let its coordinates be \\( \\left(t,-\\frac{1}{2 \\sqrt{2}} t+\\frac{5}{2}\\right) \\), where t>0\nAccording to \\( \\mathrm{RP}^{2}=(t-\\sqrt{2})^{2}+\\left(-\\frac{1}{2 \\sqrt{2}} t+\\frac{1}{2}\\right)^{2}=\\frac{9}{8}(t-\\sqrt{2})^{2} \\),\n\ \\mathrm{RP}=\\mathrm{PQ} \ implies \ \\mathrm{RP}^{2}=\\mathrm{PQ}^{2} \, hence\n\\( \\frac{9}{8}(t-\\sqrt{2})^{2}=\\left(-\\frac{1}{2 \\sqrt{2}} t+\\frac{5}{2}\\right)^{2} \\)\nOn simplification, we get \n\ t^{2}-\\sqrt{2} t-4=0 \\nSolving this gives t=2 \\sqrt{2},-\\sqrt{2} t>0, so t=2 \\sqrt{2}\nTherefore, the coordinates of point P are \2 \\sqrt{2}, \\frac{3}{2}\\n\\( \\triangle \\mathrm{PQR}=\\frac{1}{2} \\cdot \\frac{3}{2}(2 \\sqrt{2}-\\sqrt{2})=\\frac{3 \\sqrt{2}}{4} \\)'

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'In triangle ABC, let D be the point that divides side BC in the ratio 1:2. Prove that: 2AB² + AC² = 3AD² + 6BD².'

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'The region D represented by simultaneous inequalities (1) to (4) is the hatched part in the diagram. The region D includes the boundary lines. Let k be the total production quantity, then x+y=k. Inequality (5) represents a line with a slope of -1 and a y-intercept of k. To find the maximum value of k when this line (5) intersects with region D, we can determine that when the line (5) passes through the point (10,4), the value of k is maximized. In this case, k=10+4=14, therefore, the maximum total production quantity is 14 units.'

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'Prove that the three perpendiculars dropped from the three vertices of triangle ABC to the opposite sides or their extensions intersect at one point.'

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'Find the locus of a point P that is equidistant from points A(-1,-2) and B(-3,2).'

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'Prove the following inequalities. Also, determine when the equality holds.'

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'Equation of a line passing through the intersection of two lines'

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'In diagram E, the two triangles with slanted lines are congruent, so the sum of angles a and b is 90 degrees. Therefore, angle RPQ is also 90 degrees, so triangle PQR is a right isosceles triangle. Hence, the sum of angles x and y in diagram W is 45 ✕ 2 = 90 degrees.'

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'From figures 2, 3, and 4, select all the correct sentences to describe the weather in the Kanto region and answer with symbols.'

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'Figure 7 is a diagram viewed from above showing Tetsuo extending his right hand towards a mirror. While standing to align the line connecting the right eye, left eye, and the tip of the right hand H parallel to the mirror, consider observing the reflected tip of the hand H on the mirror. Mark point R on the mirror surface so that H coincides with the reflected image when looking at the mirror with only the right eye, and mark point L in a similar manner when looking at the mirror with only the left eye. (1) Draw points R and L on the diagram on the answer sheet without erasing the lines used for drawing. (2) After marking R and L on the mirror surface, move one step closer to the mirror perpendicular to the mirror (direction of the arrow) while maintaining the same posture to observe the reflected tip of the hand H. When looking at the mirror with only the right eye, how does the reflected H appear in relation to point R? Choose the most appropriate answer. Similarly, when looking at the mirror with only the left eye, how does the reflected H appear in relation to point L? Choose the most appropriate option and answer.'

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"(6) The vernier scale of the caliper in Figure 6, like (1), has a minimum interval of 1.95mm. In Figure 6, the scale line corresponding to P in Figure 3 is slightly beyond 20mm of the main scale (corresponding to position P' in Figure 3). Also, where Q corresponds to Figure 3, the vernier scale reads 3.5 and the main scale reads 34mm. By similar reasoning to (4), the length of PP' is calculated as 34 - 20 - 1.95 × 3.5 × 2 = 0.35mm. Therefore, the button diameter is determined as 20 + 0.35 = 20.35mm."

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"Choose the correct answer for the direction of the line drawn on block B and the angle at which block B has rotated, and provide the symbol. In the diagram, the dashed line represents the direction of the original block's line and the double-lined arc represents the rotation angle of block B."

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'For the following statements regarding item 11k, X and Y, choose the correct combination of true and false from the options below and answer with the corresponding number.'

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'Problem about the elongation of a metal rod and its measurement.'

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'We conducted an experiment on the generation and properties of ammonia. Answer each of the following questions.'

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'From the way the graph folds in the text, when drawing the graph of C in Fig. 3, it can be seen that it forms a thick dashed line. From Fig. 3, it can be understood that the combination of (longest, shortest) regarding the length before igniting is (B, A). Moreover, from the slope of the graph, the combination of (longest, shortest) for the length that burns in one minute is found to be (C, A). Therefore, ① is (ウ), and ② is (オ).'

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'In the diagram on the right, when a blue color is placed at position a, it is not possible to place blue at position 1 <wide> (3). Therefore, when blue is placed at position b, it cannot be placed at positions (4) to (6). Similarly, by considering further, it is possible to place blue at positions c→d→e→f, and it can be seen that a maximum of 6 blues can be placed. From this state, it is possible to move the blue at position b to position (5), and at the same time, it is also possible to move the blue at position a to position (2). In other words, out of the 4 blues in the I column, there are 3 ways to turn 2 of them blue, namely (a and b), (a and (5), (2) and (5)). The same applies to the I column and the positive column, so there are a total of 3 × 3 × 3 = 27 ways to place 6 blues. Furthermore, in all cases, there are 2 possibilities for the remaining pattern, so the total is calculated as 27 × 2 = 54 ways.'

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'To draw the planar figure 1, on the line extended from OD by a length of (1) in figure(1), take point L such that OD equals DL, draw the perpendicular bisector of OL. For this, take point M on the left side of OL such that OM equals LM, and point N on the right side of OL such that ON equals LN, and connect M and N. Next, connect points P and D, and draw the perpendicular bisector of PD. For this, take point Q on the left side of PD such that PQ equals DQ, and point R on the right side of PD such that PR equals DR, and connect Q and R. Furthermore, let the intersection of lines MN and QR be S. Finally, draw a part of a circle with center at S passing through D and P, where the intersection point of the circle with center at O, excluding D, is E.'

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'(1) Consider the cases of figures A (blue in the middle), B (blue in the corners), and C (blue in the middle of the edges). In each case, the remaining parts can only be pasted in one unique way to ensure they are not the same when rotated. In this situation, if one half is yellow and the other half is red, or if one half is red and the other half is yellow, there can be two patterns in all cases. Next, considering the placement of the blue areas, there is one possibility in the case of figure A, and in the cases of figures B and C, when rotated 90 degrees, there are 4 possibilities each. Therefore, there are a total of (1+4+4)×2=18 possibilities.'

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'Considering part d underlined, explain in 20-30 words the advantages of wooden tablets compared to paper.'

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'Find the angle between the line $\\ell: \\frac{x-2}{4}=\\frac{y+1}{-1}=z-3$ and the plane $\\alpha: x-4 y+z=0$.'

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'Prove that the incenter P(z) of triangle OAB with vertices O(0), A(α), B(β) satisfying the given equation.'

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'When the point P(z) moves along the straight line passing through -1/2 and perpendicular to the real axis, what kind of shape does the point Q(w) represented by w=1/z draw?'

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'Find the range of values for q such that the plane PQR intersects the edge OD.'

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'Find the equation of the plane passing through point A(1,1,0) and perpendicular to the line $\\frac{x-6}{3}=y-2=\\frac{1-z}{2}$.'

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'In parallelogram ABCD, let point P be the internal division of diagonal AC in the ratio 3:1, and point Q be the internal division of side BC in the ratio 2:1. Prove that points D, P, Q are collinear.'

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'When the two endpoints X and Y of a line segment of length 2 move along the x-axis and y-axis, respectively, and a point Q is taken on the extension of line segment XP such that PQ = 1, find the trajectory of point Q.'

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'Describe the conditions required to meet the collinear, concurrent, and coplanar conditions.'

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'Question: Prove the Midpoint Theorem, which states that in triangle ABC, if D and E are the midpoints of sides AB and AC respectively, then BC // DE and BC = 2DE.'

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'In polar coordinates, let point A(3, π) be on a line g that is perpendicular to the starting line. Find the polar equation of the trajectory where the ratio of distances from the pole O and line g to point P is constant. (A) 1:2 (B) 1:1'

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'Practice to prove that in triangle OAB with three different points O(0), A(α), B(β) as vertices, the incenter of vertex O as P(z), then z satisfies the following equation.'

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'Find the geometric shape represented by all points $P(z)$ that satisfy the equation $|z-\\alpha|=|z-\eta|$.'

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"Proof by elementary geometry for Example 123\n1. The strategy of extending the median to create a parallelogram can be used to prove the statement, but it requires additional points and auxiliary lines, making it less intuitive. By extending point M on segment AM such that AM = MH, and since EM = GM, it can be deduced that quadrilateral AGHE is a parallelogram. Therefore, AE = GH, and from AB = AE, we get AB = GH. Furthermore, AC = AG and AE // GH, leading to ∠EAG + ∠AGH = π, hence ∠AGH = π - ∠EAG = ∠BAC. Consequently, ∠AGH = ∠BAC (1) - (3), and from ∠AGH = ∠BAC, it follows that triangle ABC ≡ triangle GHA, so BC = AH = 2AM. Introduce point B' such that BC // B'A, which implies ∠MAE = ∠GHA = ∠ABC = ∠BAB'. Thus, ∠MAB' = ∠MAE + ∠EAB' = ∠BAB' + ∠EAB' = π/2, showing that AE // GH and the angles between two sides are equal. Therefore, AM ⊥ B'A and BC // B'A leads to AM ⊥ BC and BC // B'A."

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'Basic Example 121 Application of the Cosine Rule'

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'Please explain the converse of the Pythagorean theorem.'

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'Prove that the equation (1 + tan^2(A/2))sin^2((B+C)/2) = 1 holds true when the sizes of angles A, B, and C of triangle ABC are represented by A, B, and C respectively.'

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"From the result of (2), it can be observed that since point X divides the side AC externally in a specific ratio, and point Y also divides the side AC externally in an equal ratio, points X and Y coincide. Therefore, we can conclude that the three lines AC, PQ, and RS intersect at a single point.\n\nLet's confirm this using a geometric drawing software. Regardless of the changes made to the shape of quadrilateral ABCD, the three lines AC, PQ, and RS will intersect at a single point. Please move the positions of points P, Q, R, and S in the software."

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'What is the equation of the parabola obtained by translating the parabola y=2x^{2} parallel to the x-axis by -2 units and parallel to the y-axis by 3 units?'

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'In tetrahedron ABCD with BC=BD, let AO be the perpendicular from point A to plane BCD. If point O lies on the angle bisector of ∠CBD at point E, prove that AE is perpendicular to CD.'

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'Proof: For any point P not on the triangle ABC and its sides and medians, the following equation holds: \AP^{2}+BP^{2}+CP^{2}=AG^{2}+BG^{2}+CG^{2}+3 GP^{2}\'

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"Regarding the semi-line OX at 80 degrees, symmetrical to point A is A', and regarding the semi-line OY, symmetrical to point B is B'. Let the intersection of line A'B' and semi-line OX be P, and the intersection of line A'B' and semi-line OY be Q."

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Join our Discord#### Q.82

'Prove that in triangle ABC, if O is the circumcenter and P, Q, R are the symmetrical points of sides BC, CA, AB respectively, then O is the orthocenter of triangle PQR.'

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'Consider a cube as shown in the figure, draw evenly spaced lines on the adjacent three faces ABCD, BEFC, and CFGD, and assume that it is possible to move along these lines. In each of the following cases, determine the number of possible shortest path routes: (1) Going from A to C on the face ABCD (2) Going from A to F on the faces ABCD and BEFC (3) Going from A to F on the faces ABCD, BEFC, and CFGD'

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'In the diagram on the right, when using red, blue, yellow, and white paints to clearly delineate A, B, C, and D'

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Join our Discord#### Q.85

'There are two circles tangent at point A. When a tangent at point B on one circle intersects the other circle at points C and D, prove that AB bisects the exterior angle of ∠CAD.'

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Join our Discord#### Q.86

'Prove that the following equations hold in triangle ABC.'

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Join our Discord#### Q.90

'Given three points A, Q, B on the circumference of a circle, and a point P on the line AB such that P is on the same side of Q, prove the following proposition using proof by contradiction: ∠APB > ∠AQB ⇒ Point P is inside the circle.'

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Join our Discord#### Q.91

'Construct a square PQRS based on the vertices of triangle ABC.'

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Join our Discord#### Q.92

'Prove that the angle bisector at point A, the 60-degree angle bisector at point B, and the angle bisector at point C in triangle ABC intersect at a single point.'

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Join our Discord#### Q.93

'The line AG is the angle bisector of the exterior angle BAC. Let H be the point on the extension of the ray BA, prove the following equation:'

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Join our Discord#### Q.94

'Given line segment AB and point P on it. Construct a right triangle ABC with AB as the hypotenuse, take point Q on segment AC, and point R on segment BC such that quadrilateral PQCR becomes a square. Draw the square PQCR.'

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'In the figure, let E be the point where the angle bisector of the exterior angle CAD of triangle ABC, which is inscribed in a circle, intersects the circle again, and let F be the point where it intersects the extension of side BC. If AE = AC, prove that BE = CF.'

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Join our Discord#### Q.96

'Proof problem about the equation a^2 + b^2 = c^2'

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Join our Discord#### Q.97

'In trapezoid ABCD, where PR / / BC, PR ≠ BC, let points P and Q divide the sides PR and BC in ratio m: n. Prove that the lines AB, CD, and PQ intersect at one point.'

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'In tetrahedron ABCD with BC=BD as shown in the figure on the right, let AO be the perpendicular from point A to plane BCD. If point O lies on the angle bisector BE of angle CBD, prove that AE is perpendicular to CD.'

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'Prove that, as shown in the diagram on the right, for a right triangle ABC where ∠B=90 degrees, a point D is taken on the side BC (with D different from B and C). Then, a point E is taken such that ∠ADE=90 degrees and ∠DAE=∠BAC. Prove that the four points A, D, C, E lie on the same circle.'

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'Prove that the angle bisector of angle A in triangle ABC intersects with side BC at point D, dividing BC internally in the ratio of AB:AC. Prove this in the following two ways:\n(1) Focus on triangles ABD and ECD when a line parallel to AB passing through point C intersects with line AD.\n(2) Focus on the areas of triangles ABD and ACD by dropping perpendiculars from point D to lines AB and AC.'

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'Concerning the choices other than 1 and 3 in reference (5). First, as the points D, A, P are on the same line, they cannot be on the same circle. Therefore, options containing points A and P are not applicable. Next, according to the answers, the four points D, A, C, E are on the same circle (top-right diagram). Hence, the circumcircle of triangle DAE must pass through point C, so it will not pass through point F. Therefore, option 3 is not applicable. Similarly, the circumcircle of triangle DCE must pass through point A, so it will not pass through point F. Therefore, option 4 is not applicable. Similarly, by considering the circle passing through the 4 points D, C, P, Q (bottom-right diagram), it is evident that option 5 is also not applicable.'

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'Construct a circle that intersects with ray a at point A on ray a, and cuts out a line segment of length l from ray b starting from point O.'

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'Prove that the following equation holds for any point P not on the sides, medians, or extensions of triangle ABC, with its centroid denoted as G:'

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'Prove that in triangle ABC, when the circumcenter is O and the symmetrical points with respect to sides BC, CA, AB are P, Q, R respectively, O is the orthocenter of triangle PQR.'

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'Prove that in triangle ABC with ∠B=90°, when point P is on side BC, we have AB < AP < AC.'

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'Basic Example 67 Circumcenter and Orthocenter of a Triangle'

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'Want to color the regions A, B, C, D, E in the diagram on the right. Different colors should be used for adjacent regions, and the specified number of colors must all be used. How many ways are there to color? (1) Using 5 colors (2) Using 4 colors (3) Using 3 colors'

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"Using Thales' theorem, prove that if the circumcircle of triangle ABC intersects the angle bisector of ∠BAC at point M, then MA = MB + MC implies AB + AC = 2BC."

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'As shown in the figure on the right, three points D, E, F are taken outside triangle ABC such that triangles ABD, BCE, and CAF each form an equilateral triangle.'

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'In trapezoid ABCD where AD // BC, let points P and Q be where the sides BC and DA are internally divided in the ratio of m:n. Prove that the lines AC, BD, and PQ intersect at a single point.'

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'Given a right triangle ABC with ∠A=90°, construct equilateral triangles BAD and ACE on the exterior. Let the intersection of segments CD and BE be P. Prove that points C, E, A, P lie on the same circle.'

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'Find the total number of paths that satisfy the following conditions for the shortest path from point P to point Q on the right side of the PR diagram:'

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"In the given problem, solve using the properties of centroid and the relationship with medians. (1) Focus on two medians and show their relationship to the centroid. (2) Use the centroid property of '2:1 ratio division' to find the ratio of areas."

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"Using Menelaus' theorem, prove the following when the sides BC, CA, AB of triangle ABC or their extensions intersect a line l not passing through the vertices of the triangle at points P, Q, R respectively."

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"Let's think about a problem related to the properties of geometric shapes. By using the properties of the following shapes, we will provide a proof for the specified problem."

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'Basic Example 82 The Inverse of the Circle Theorem\nIn the diagram to the right, L, M, N are the midpoints of the sides \ \\mathrm{AB}, \\mathrm{BC}, \\mathrm{AD} \ of the quadrilateral \ \\mathrm{ABCD} \ inscribed in a circle. Moreover, the intersection of line ML and line DA is P, and the intersection of line NL and line CB is Q. Prove that these 4 points M, N, P, Q are on the same circumference.'

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'In triangle ABC, if the incenter I is symmetric with respect to the sides BC, CA, and AB and denoted as P, Q, R respectively, what type of point is I with respect to triangle PQR?'

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'Perpendicular bisector: Point P lies on the perpendicular bisector of line segment AB. \ \\Leftrightarrow \ Point P is equidistant from points A and B.'

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'Prove that the lengths of the two tangents drawn to a circle from a point outside the circle are equal.'

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Join our Discord#### Q.24

'Prove that triangle ABC is an equilateral triangle if the centroid and orthocenter coincide.'

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'Triangle ABC is given as shown on the right side. Construct a square PQRS such that side QR lies on segment BC, vertex P lies on segment AB, and vertex S lies on segment AC.'

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'Prove that the line AB is a tangent to the circumcircle of triangle BDF.'

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Join our Discord#### Q.27

'Prove that the three lines AB, CD, PQ intersect at one point. Hint: Show that the intersection of AB and PQ is the same as the intersection of CD and PQ.'

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'The diagram on the right shows a box plot of the scores obtained by 30 students in a science test. When the scores on which this box plot was based are plotted as a histogram, which of the following corresponds to it?'

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'As shown in the diagram on the right, draw perpendiculars PD, PE, PF from point P on the circumcircle of triangle ABC to lines AB, BC, CA, respectively. Prove the following.'

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Join our Discord#### Q.30

'Using the theorem of inscribed angles, prove the condition for points A, B, P, Q to lie on a single circle.'

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Join our Discord#### Q.31

'Angle bisector: Point P lies on the angle bisector of angle ABC.'

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Join our Discord#### Q.32

'Proof of quadrilateral inscribed in a circle - Basic Example 83'

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'Practice 68\nLet the orthocenter of acute triangle ABC be H, the circumcenter be O, the midpoint of side BC be M, and the midpoint of segment AH be N. Prove that the length of segment MN is equal to the radius of the circumcircle of triangle ABC using the fact that AH=2OM.'

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Join our Discord#### Q.34

"Using the Thales' theorem, prove that if the circumcircle of triangle ABC intersects the angle bisector of ∠BAC at M, then when MA = MB + MC, we have 84AB + AC = 2BC."

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Join our Discord#### Q.35

'Prove that the following equation holds in triangle ABC with centroid G: AB² + BC² + CA² = 3(AG² + BG² + CG²)'

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'In an isosceles triangle ABC where AB=AC, take two points F and G on the base BC, draw the chords AFD and AGE of the circumscribed circle of triangle ABC. Prove the following: (1) AB² = AF * AD (2) The four points D, E, F, G are concyclic.'

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Join our Discord#### Q.37

'Prove that in triangle ABC, when the midpoint of side BC is M, the equation AB²+AC²=2(AM²+BM²) holds true (Median Theorem).'

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Join our Discord#### Q.38

'Prove that in trapezoid ABCD, where AD//BC, AD≠BC, the sides AD and BC are divided into points P and Q in the same ratio of m:n. Then prove that the lines AB, CD, and PQ will intersect at one point.'

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'Draw a sector OAB with O as the center as shown on the right. On the line segment OA, draw a square PQRS where the side QR coincides with OA, and the vertex P lies on the line segment OB, while the vertex S lies on the arc AB.'

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"Prove using the converse of Ceva's theorem that the angle bisectors of the three interior angles of a triangle intersect at one point."

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Join our Discord#### Q.45

'On circle O, draw tangents PA and PB from an external point P, and draw chord CD passing through point M where line segment AB intersects line segment PO. Prove that points P, C, O, and D lie on the same circle. Here, C and D are not on the line PO.'

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'Given points A and M inside and on the circumference of circle O. Now, draw a chord PQ passing through M such that AM bisects ∠PAQ.'

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"Prove that if the two circles O and O' passing through a point P have common external tangents C and D, and if the line passing through point P intersects the two circles at points A and B, then AC is perpendicular to BD."

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'Let O be a given circle with fixed points A and M inside. Draw a chord PQ passing through M such that AM bisects ∠PAQ. Construct such a chord PQ.'

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'Master the use of Venn diagrams to conquer Example 49!'

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'Given line segment AB with a length of 1 and line segments with lengths of a and b, draw a line segment with a length of b/3a.'

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Join our Discord#### Q.51

'Prove that the orthocenter of triangle ABC and the circumcenter of triangle PQR coincide when the midpoints of the sides QR, RP, and PQ of acute triangle PQR are denoted by A, B, and C respectively. Let AD, BE, and CF be the altitudes drawn from vertices A, B, and C of triangle ABC to the opposite sides.'

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'Draw a rectangle PQRS inside the acute angled triangle ABC as shown on the right, such that 2PQ=QR, side QR lies on side BC, vertex P lies on side AB, and vertex S lies on side CA. (Describe only the method of drawing)'

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Join our Discord#### Q.54

'In triangle ABC where AB=5, BC=6, CA=4 and point P is the intersection of the bisectors of the exterior angles of angles B and C, prove that line AP bisects angle A.'

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Join our Discord#### Q.55

'Translate the given text into multiple languages.'

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'In the figure to the right, △ABC and △CDE are both equilateral triangles, and the vertices B, C, D are collinear. Let F be the intersection of AD and BE, prove that points A, B, C, F are concyclic.'

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'Let\'s learn about proof methods such as contraposition and proof by contradiction. Please prove the following proposition by proof by contradiction: "When dividing 12 candies among 3 people (A, B, C), at least one person will receive 4 or more candies."'

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Join our Discord#### Q.58

'Want to color the regions A, B, C, D, E in the right figure. When color adjacent areas with different colors, how many ways are there to color with three colors?'

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Join our Discord#### Q.59

'Problem: Length in proportion, represented as the product of segments\nGiven a segment AB of length 1 and segments of length a, b:\n(1) Draw a segment of length a/b.\n(2) Draw a segment of length 2ab.'

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"Let's review the length of the tangent and the tangent theorem!"

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Join our Discord#### Q.61

'Prove that in an acute triangle ABC, where the perpendicular from vertices B and C to the opposite sides are BE and CF respectively, with their intersection point as H. Let the intersection point of line AH and side BC be D, prove that AD is perpendicular to BC.'

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'Quadrilateral ABCD is inscribed in a circle, with AB=4, BC=2, and DA=DC. Let E be the intersection point of the diagonals AC and BD, F be the point that divides segment AD in a 2:3 ratio, and G be the intersection point of line FE and DC. (2) Consider the case when line AB passes through point G. In this case, as point B lies on side AG of triangle AGD, we have BG=. Also, since line AB intersects line DC at point G, and the 4 points A, B, C, D lie on the same circumference, we have DC=.'

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Join our Discord#### Q.63

"Prove that in triangle ABC, the angle bisectors of angle B and angle C intersect at point I. Let's draw perpendicular lines from point I to sides BC, CA, and AB, denoted as IP, IQ, IR, respectively. We have IR=IP, IP=IQ, therefore IR=IQ, which means IP=IQ=IR. Hence, point I lies on the angle bisector of angle A. Therefore, the angle bisectors of the three interior angles of triangle ABC intersect at one point I. This point where the angle bisectors intersect is called the incenter of the triangle, and the circle with the incenter as its center and tangent to the three sides is called the incircle."

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'Draw a square PQRS inside the sector OAB with center O, such that side QR lies on segment OA, vertex P lies on segment OB, and vertex S lies on arc AB, according to the diagram on the right. (Provide only the method of drawing).'

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Join our Discord#### Q.66

'Draw the point that divides the given line segment AB internally in the ratio 3:2.'

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Join our Discord#### Q.67

'Draw the perpendicular bisector of line segment AB (1) Draw two circles with points A and B as centers, each with equal radius, and mark the intersection points of the two circles as C and D. (2) Draw the line CD.'

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Join our Discord#### Q.68

'For (1) and (2), find the length of segment AD according to the instructions.'

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'As shown in the figure to the right, two circles with different radii are tangent at point A. Draw a line tangent to the inner circle at point D, and let B and C be the intersection points with the outer circle. Prove that AD bisects ∠BAC.'

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Join our Discord#### Q.70

'Given example problem of segregating 25 regions (planar):'

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Join our Discord#### Q.71

'For subsets A, B of the universal set TR, verify that the following equation holds using diagrams.'

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'In triangle ABC with its circumcircle, point D is taken on side BC such that ∠BAD=∠CAD. Furthermore, let P be the intersection of the tangent to the circle at point A and line BC. Prove that PA=PD.'

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Join our Discord#### Q.73

'Draw a square PQRS inside the sector OAB with O as the center, such that side QR lies on line OA, vertex P lies on line OB, and vertex S lies on arc AB (provide only the method of drawing).'

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Join our Discord#### Q.74

'I want to color the areas A, B, C, D, and E on the right diagram. When coloring adjacent areas with different colors using no more than 4 colors, how many ways are there to color the areas?'

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Join our Discord#### Q.75

"Prove that when two intersecting circles O and O' have a common chord AB passing through point P, the chord of circle O passing through P is CD and the chord of circle O' passing through P is EF, then the four points C, D, E, F lie on the same circumference. However, note that the four points C, D, E, F are not collinear."

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Join our Discord#### Q.76

'Prove the following in tetrahedron ABCD: (a) Let M be the midpoint of edge AD, then AD is perpendicular to plane MBC; (b) AD is perpendicular to BC.'

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'There is a line ℓ on plane α. Point A is not on α, point B is on ℓ, and point O is on α but not on ℓ. Prove the following: OB is perpendicular to ℓ, AB is perpendicular to ℓ, OA is perpendicular to OB implies OA is perpendicular to α.'

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"There are two circles O and O' intersecting at points P and Q. The line passing through point P intersects circles O and O' at points A and B, and the line passing through points A and Q intersects circle O' at point C. Let AD be the tangent to circle O at point A, then prove that AD // BC."

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Join our Discord#### Q.79

'From point P, drop perpendiculars to lines BC and AC, and let the intersections be E and F respectively. By using the congruences of triangles PBD ≡ PBE, PCE ≡ PCF, and PAD ≡ PAF, express AD in terms of AB, BC, and CA.'

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Join our Discord#### Q.80

'In the figure on the right, find x, y, z. Here, ℓ is the tangent to circle O, and point A is the point of contact. Also, in (2), ∠ABD=∠CBD.'

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Join our Discord#### Q.81

"In the figure on the right, there are two circles O and O' intersecting at points P and Q. Let A be the intersection of the tangent line to circle O at point P and circle O', B be the intersection of line AQ and circle O, and C be the intersection of line BP and circle O'. Prove that AC=AP."

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Join our Discord#### Q.82

"Given two circles O and O' intersect at two points P and Q as shown in the diagram. Let's denote the tangent from point P to circle O intersecting circle O' at point A, the intersection of line AQ and circle O at point B, and the intersection of line BP and circle O' at point C. Prove that AC = AP."

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Join our Discord#### Q.84

'Given three points A, B, Q on the circumference of a circle, and a point P such that it lies on the same side of line AB as point Q, prove the following statements: \n1. If point P is on the circle ⇒ ∠APB = ∠AQB \n2. If point P is inside the circle ⇒ ∠APB > ∠AQB \n3. If point P is outside the circle ⇒ ∠APB < ∠AQB'

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Join our Discord#### Q.85

'Constructing the bisector of angle AOB (1) Draw a circle with center O and a suitable radius, and mark the intersections with the half-lines OA and OB as C and D, respectively. (2) Draw circles with centers at points C and D and equal radii, and mark one of the intersection points of the two circles as E. (3) Draw the half-line OE.'

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Join our Discord#### Q.86

'In the given figure, find the values of α and β. Where ℓ is the tangent to circle O, and point A is the point of tangency. Also, in (3) it is given that PQ // CB.\n(1)\n(2)\n(3)\nFocus on the tangent and the triangle using the tangent theorem.\n(2) The measure of ∠CAB can be found using the inscribed angle theorem.\n(3) From PQ // CB, ∠ABC=∠BAQ\nIn (1), let point D be on line ℓ as shown in the figure\n\n∠BAD =∠OAD-∠OAB\n=90°-20°=70°\n\nTherefore, α=∠BAD=70°'

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Join our Discord#### Q.87

'By drawing a perpendicular line to PQ after drawing parallel lines (*), a line parallel to l can be created.'

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Join our Discord#### Q.89

'Constructing the perpendicular line of line segment PQ: (1) Draw a circle with center at point P and any radius, intersecting with line l at points A and B. (2) Draw circles with centers at points A and B, each with equal radius, and define one of the intersection points of these two circles as Q. (3) Draw the line PQ.'

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Join our Discord#### Q.90

"Prove the following using the converse of the Ceva's theorem: (1) The three medians of a triangle intersect at one point. (2) The three angle bisectors of a triangle intersect at one point."

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Join our Discord#### Q.91

'As shown in the figure on the right, two circles with different radii are tangent to point A. Draw a line passing through point D on the inner circle, with points B and C as the intersection points with the outer circle. Prove that AD bisects angle BAC.'

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Join our Discord#### Q.92

'Let P(2,3,1) be a point. Let D, E, and F be points symmetric to P with respect to the xy plane, yz plane, and zx plane, respectively. Determine the coordinates of points D, E, and F.'

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Join our Discord#### Q.93

'Prove that the arc length is equal for equal central angles in a circle.'

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Join our Discord#### Q.94

'Draw the given line segment AB and mark the point that divides it externally in the ratio 5:1.'

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Join our Discord#### Q.95

'There are two circles that are tangent to point P. As shown in the diagram on the right, if two lines passing through point P intersect with the outer circle at points A, B and with the inner circle at points C, D. Prove that AB and CD are parallel.'

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Join our Discord#### Q.96

'(1) Which of the rectangles on the right, ABCD, is circumscribed by a circle?\n(2) In acute triangle ABC, on edge BC, point D (different from points B and C) is taken, and perpendiculars DE, DF are drawn from point D to edges AB and AC, respectively. Prove that quadrilateral AEDF is inscribed in a circle.'

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Join our Discord#### Q.97

'Prove Theorem 17: If two line segments AB and CD, or the extensions of AB and CD intersect at point P, and PA * PB = PC * PD, then points A, B, C, D lie on the same circumference.'

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Join our Discord#### Q.99

'In the diagram on the right, since ∠CAD=∠EBC, quadrilateral ABDE is circumscribed about the circle. Therefore, by the theorem of inscribed angles, ∠ADE=∠ABE. Also, since ∠BEC=90 degrees, ∠ADC=90 degrees, quadrilateral CEHD is circumscribed about the circle. ∠HEC + ∠HDC = 180 degrees. As the sum of opposite angles is 180 degrees, quadrilateral CEHD is circumscribed about the circle.'

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'Prove that for any three distinct lines l, m, n, if l // m and m // n, then l // n.'

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'In triangle ABC, if the midpoint of side AB is D and the midpoint of side AC is E, then DE // BC and DE = 1/2 BC. The line segments connecting the vertices of the triangle to the midpoints of the opposite sides are called medians of the triangle. Looking at the three medians of a triangle, there is the following property. Theorem 5: The three medians of a triangle intersect at a single point, which divides each median in a 2:1 ratio.'

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'Plot the point that divides the given line segment AB internally in the ratio 1:4.'

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'Prove that in a scalene triangle ABC, with the incenter I and lines BI, CI intersecting sides AC, AB at points E, D, respectively, if DE // BC, then AB=AC.'

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'Chapter 3 Figures and Equations 97 (2) Line BC is taken on the x-axis, and a line perpendicular to line BC passing through point D is taken on the y-axis, where point D becomes the origin O, which can be represented as A(a, b), B(-3c, 0), C(2c, 0). In this case, 2AB²+3AC² = 2{(-3c-a)²+(-b)²}+3{(2c-a)²+(-b)²} = 5a²+5b²+30c² = 5(a²+b²+6c²) Also, 3AD²+2BD² = 3{(-a)²+(-b)²}+2(3c)² = 3(a²+b²+6c²) ...(2) From (1) and (2), we have 3(2AB²+3AC²)=5(3AD²+2BD²)'

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'Summarized some theorems, formulas, and important properties.'

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'How can the shaded part of the figure be represented by inequalities? Show it step by step.'

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'In the xy-plane, there are three points A(2, -2), B(5, 7), C(6, 0). Prove that the perpendicular bisectors of each side of triangle ABC intersect at one point (this intersection point is the center of the circumcircle of triangle ABC, also known as the circumcenter).'

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'On a plane, there are three points A(2,-2), B(5,7), C(6,0). Prove that the perpendicular bisectors of each side of triangle ABC intersect at one point (this intersection point is the circumcenter of triangle ABC, also known as the circumcenter). HINT: Prove that the intersection point of the perpendicular bisectors of segment AC and segment AB lies on the perpendicular bisector of segment BC.'

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'Find the equation of the line passing through the origin, where the perpendicular distance from the line to the points (5,0) and (3,6) are equal.'

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'Plot the region represented by the following inequalities.'

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'Find the equation of the line passing through the origin when the perpendicular distances from the points (5,0) and (3,6) to line l are equal.'

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'Prove: In triangle ABC, the midpoint of side BC is M, then AB^2 + AC^2 = 2(AM^2 + BM^2) (Median theorem).'

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'Prove that the centroids of triangle DEF and triangle ABC coincide when points D, E, F are taken on the sides BC, CA, AB of triangle ABC such that BD:DC=CE:EA=AF:FB.'

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'How to find the equation of a tangent to a circle'

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'Chapter 3 Shapes and Equations\nThe condition for two lines (1), (2) to be perpendicular is\n-3⋅(-\\frac{1}{a})=-1, solving for a, we get a=\\uparrow-3\nAnother way, the condition for two lines (1), (2) to be parallel is\n\3⋅a-1⋅1=0, hence a=\\frac{1}{3}\\nThe condition for two lines (1), (2) to be perpendicular is\n\3⋅1+1⋅a=0, hence a=\\uparrow-3\\nPerpendicular ⇔ Product of slopes is -1 ⇽ 2 lines\na_{1} x+b_{1} y+c_{1}=0 and\na_{2} x+b_{2} y+c_{2}=0\nParallel ⇔ a_{1} b_{2}-a_{2} b_{1}=0\nPerpendicular ⇔ a_{1} a_{2}+b_{1} b_{2}=0'

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'Prove that in triangle ABC, when G is the centroid, the equation AB^2 + BC^2 + CA^2 = 3(GA^2 + GB^2 + GC^2) holds true.'

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'Chapter 3 Geometry and Equations\n121\nConsider the line 2x - y + 3 = 0, let Q be a point and P be its symmetric point. Find the locus of point P as point Q moves along the line 3x + y - 1 = 0.'

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'Determine the value of the positive constant a so that the areas enclosed by the curves y=x^{3}-(2 a+1) x^{2}+a(a+1) x and y=x^{2}-a x are equal.'

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'Find the coordinates of point Q after rotating point P(4,2) around point A(2,5) by π/3.'

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'Line passing through intersection point of two lines'

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'Find the area of the triangle formed by the lines x - y + 1 = 0, 2x + y - 2 = 0, x + 2y = 0.'

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'The relationship between the lengths of the sides of a triangle and trigonometric functions is as follows.'

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'Prove the equality that shows the sizes of the internal angles A, B, and C of triangle ABC.'

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'Please indicate the related pages for the following terms: ① condition ② transitive law (inequality) ③ tangent'

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'Prove that the equation (1+tan²(A/2))sin²((B+C)/2)=1 holds when the interior angles ∠A, ∠B, ∠C of the triangle ABC are denoted as A, B, C respectively.'

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'As shown in the diagram on the right, a regular dodecagon is divided into 12 congruent triangles by diagonals. Taking points O, A, B, we have ∠AOB=360°÷12=30°, OA=OB=a. Applying the cosine rule to triangle OAB, we get 1=(2-√3)a², hence a²=1/(2-√3)=(2+√3)/((2-√3)(2+√3))=2+√3. Therefore, S=12 triangles OAB=12×1/2a²sin30°=3(2+√3)'

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'(2) moved parallel to x-axis by 1, axis is the line x=1, vertex is at point (1,0)'

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'In triangle ABC, when the following equations hold true, what kind of triangle is it? (1) a sin A + c sin C = b sin B (2) b cos B = c cos C Important 160'

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'When summarizing the solution to triangles, out of the six elements of a triangle (3 sides a, b, c and 3 angles A, B, C), in order to uniquely determine a triangle, at least one of the following three elements containing at least one side is necessary as a condition: [1] one side and its adjacent angles [2] two sides and the angle between them [3] three sides. Based on these conditions, when determining the other three elements, we explain the usage of the theorem according to the conditions.'

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'In triangle ABC, prove the following equation holds: \n\\[\\left(b^{2}+c^{2}-a^{2}\\right) \\tan A=\\left(c^{2}+a^{2}-b^{2}\\right) \\tan B\\]'

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'Think of the development diagram from the side. The length of the arc of the sector is equal to the circumference of the base, so the central angle of the sector can be determined. The shortest path on the lateral surface of a spatial figure becomes a line segment connecting two points on the development diagram.'

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'There is a piece of paper in the shape of an equilateral triangle with a side length of 10 cm. Let the vertices of this equilateral triangle be labeled as A, B, C, and let point P lie on side BC such that BP = 2 cm. When folding this equilateral triangle paper so that vertex A coincides with point P, and the intersection of the fold with sides AB, AC are labeled as D, E respectively. At this time, AD = 2 square cm, AE = 1 square cm, and the area of triangle ADE is 3 square cm.'

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'Two lines can be drawn that pass through the origin and make a 15-degree angle with the line y=x. Find the equations of these lines.'

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'When symmetrically moved about the origin, the vertex becomes \\( \\left(-\\frac{3}{4}, \\frac{31}{8}\\right) \\) and forms a concave parabola\n\\[ y=2\\left(x+\\frac{3}{4}\\right)^{2}+\\frac{31}{8} \\quad\\left(y=2 x^{2}+3 x+5 \\text { is also acceptable }\\right)\\]\n\ \\hookleftarrow x \ and \ y \ coordinates are both reversed in sign, changing from concave up to concave down.'

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'Find the equation of the parabola after shifting the parabola y = -2x^2 + 3 parallel to the x-axis by -2 and parallel to the y-axis by 1.'

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'In right triangle ABC, with BC=18 and CA=6, point D is taken on the hypotenuse AB. Perpendiculars DE and DF are drawn from D to BC and CA, respectively. Find the length of segment DE and the area when the sum of the areas of triangles ADF and DBE is minimized.'

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'Let the polar coordinates of point A be (3,0). Find the polar equation of the locus of points P where the distance to the pole O is equal to the distance to the line l passing through point A and perpendicular to the initial line.'

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'In the tetrahedron ABCD with side length 1, let the midpoints of the edges AB and CD be E and F respectively, and let the centroid of triangle BCD be denoted as G.'

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'Prove that for an ellipse passing through the foci and having a chord AB parallel to the minor axis, the square of the minor axis length is equal to the product of the major axis length and the length of chord AB, which is 120. HINT: Consider the equation of the ellipse as x²/a² + y²/b² = 1 (a > b > 0).'

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'On the coordinate plane, when the length of line segment AB is 9 and its endpoints A, B move along the x-axis and y-axis respectively, find the locus of the point P which divides the line segment AB in the ratio 1:2.'

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'Prove that the midpoints of the edges AB, BC, CD, DA of quadrilateral ABCD are P, Q, R, S respectively, and the midpoints of the diagonals AC, BD are T, U. Show that the midpoints of the segments PR, QS, and TU are all the same.'

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'Prove that in tetrahedron OABC, if the centroid of ∆OAB is G1 and the centroid of ∆OBC is G2, then G1G2 is parallel to AC.'

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'Extensions of Cycloid\nCurves related to cycloids (מ.248) include the following:\n- Trochoid\nWhen a circle of radius \ a \ rotates without slipping along the fixed straight line \ (x \ axis \\), the curve traced by a fixed point \ \\mathrm{P} \ at distance \ b \ from the center of the circle is called a trochoid. In particular, when \ a = b \, the point \ \\mathrm{P} \ lies on the circumference of the circle, and the curve traced by point \ P \ is a cycloid. The parametric representation of a trochoid is given by \ x = a \\theta - b \\sin \\theta, y = a - b \\cos \\theta \ \ \\qquad \ (＊). From (＊), for example, by focusing on the right-angled triangle \ APB \ as \\( \\mathrm{P}(x, y) \\) in the above figure, one can derive \\( x = a \\theta - b \\cos \\left(\\theta-\\frac{\\pi}{2}\\right), y = a + b \\sin \\left(\\theta-\\frac{\\pi}{2}\\right) \\).\n\nRefer to epicycloids and hypocycloids on page D.253.\n\nWhen a circle \ C \ of radius \ b \ rotates without slipping while externally tangent to a fixed circle of radius \ a \, the curve traced by a fixed point \ \\mathrm{P} \ on circle \ C \ is called an epicycloid. Also, when a circle \ C \ of radius \ b \ rotates without slipping while internally tangent to a fixed circle of radius \ a \, the curve traced by a fixed point \ \\mathrm{P} \ on circle \ C \ is called a hypocycloid. As learned on the previous page, the parametric representations of these curves are as follows:\n- Epicycloid\n\\[\n\\left\\{\egin{array}{l}\nx = (a + b) \\cos \\theta - b \\cos \\frac{a + b}{b} \\theta \\\\\ny = (a + b) \\sin \\theta - b \\sin \\frac{a + b}{b} \\theta\n\\end{array}\\right.\n\\]\n\nFor example, the general shapes of epicycloids when \ a = b, a = 2b \ are as follows.\n When \ a = b \\n When \ a = 2b \\n\nNote: When \ a = b \, this curve is called a cardioid.\n- Hypocycloid\n\\[\n\\left\\{\egin{array}{l}\nx = (a - b) \\cos \\theta + b \\cos \\frac{a - b}{b} \\theta \\\\\ny = (a - b) \\sin \\theta - b \\sin \\frac{a - b}{b} \\theta\n\\end{array}\\right.\n\\]\n\nFor example, the general shapes of hypocycloids when \ a = 3b, a = 4b \ are as follows.\n When \ a = 3b \\n When \ a = 4b \\n\nNote: When \ a = 4b \, this curve is called an astroid or hypocycloid.'

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'Exercise 67 -> Booklet p.134\n(1) Let the equations of two planes α and β be (1) and (2) respectively. Subtracting (1) from (2) gives'

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'Existence range of points on a plane\nFor triangle OAB, when \ \\overrightarrow{OP} = s \\overrightarrow{OA} + t \\overrightarrow{OB} \, the existence range of point P is\n(1) Line AB \ \\Leftrightarrow s + t = 1 \\nIn particular, line segment AB \ \\Leftrightarrow s + t = 1, s \\geqq 0, t \\geqq 0 \\n(2) Circumference and interior of triangle OAB\n\ \\Leftrightarrow 0 \\leqq s + t \\leqq 1, \\quad s \\geqq 0, \\quad t \\geqq 0 \\n(3) Circumference and interior of parallelogram OACB\n\ \\Leftrightarrow 0 \\leqq s \\leqq 1, \\quad 0 \\leqq t \\leqq 1 \'

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'(1) Point F is symmetric to point C with respect to line segment OA, therefore triangle ADF ≡ triangle ADC. Hence, triangle ADF = 1/6 triangle OAB implies triangle ADC = 1/6 triangle OAB. Also, since triangle ADC = 1/3(1-α) triangle OAB, solving for α gives α = 1/2, which satisfies 0 < α < 1.'

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'Find the equation of the plane passing through the following 3 points.'

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'Solve the following vector problem: Let the areas of \ \\triangle \\mathrm{PBC}, \\triangle \\mathrm{PCA}, \\triangle \\mathrm{PAB} \ be denoted by S, where \ \\triangle \\mathrm{ABC} \ represents the area.'

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'Find the equation of a hyperbola with asymptotes as the two lines y=√3x, y=-√3x and a distance of 4 between the two foci.'

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'Important Example 57 | Equation of a Plane\nFind the equation of the plane passing through points A(0,1,-1), B(4,-1,-1), C(3,2,1).'

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'Prove that the following theorem holds:\nTheorem 1: The fractional linear transformation transforms a circle in the complex plane into a circle.'

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'Consider the positive integer 42k and two curves defined on the interval 2kπ≤x≤(2k+1)π: C₁: y=cos x and C₂: y=(1-x²)/(1+x²).'

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'Find the equation of the tangent line at point P, determine the asymptotes and coordinates of their intersection, and prove that the area of triangle OQR is independent of the choice of point P.'

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'Find the polar equation of the line with polar coordinates (p, α) of the foot H of the perpendicular dropped from the pole O to the line.'

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'Important Example 67 Equations of the intersection of planes, including the equation of the plane. Let the intersection of planes be ℓ with equations (1) α: 3x-2y+6z-6=0 ⋯ ⋯ (1) β: 3x+4y-3z+12=0 ⋯ ⋯ (2). Express the equation of intersection ℓ in the form x-x₁/l=y-y₁/m=z-z₁/n. (2) Determine the equation of the plane γ that includes the line ℓ and passes through the point P(1,-9,2).'

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"Prove the following theorem using the complex plane. For a quadrilateral ABCD inscribed in a circle of radius 100, the equation AB⋅CD+AD⋅BC=AC⋅BD holds (Ptolemy's theorem)."

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'Practice (2) Prove that when two chords PQ and RS passing through one of the foci F of a conic intersect at right angles, then 1/PF*QF + 1/RF*SF is constant.'

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'Let the part of the hyperbola x^2 - y^2 = 2 in the fourth quadrant be denoted as C, with the point (√2, 0) as A and the origin as O. Let the tangent line at point Q on the curve C and the line passing through point O perpendicular to the tangent line be denoted as P. (1) Show that as point Q moves along the curve C, the locus of point P is represented by the polar equation r^2 = 2cos 2θ (r> 0, 0 < θ < π / 4).(2) In (1), find the Cartesian coordinates of point P that maximizes the area of triangle OAP.'

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'There is a concave mirror in the shape of a hemisphere. Let O be the center of the sphere, r be the radius, and AB be a diameter. If a ray of light forms an angle θ with the diameter AB starting from point A and reflects at point P on the mirror, intersecting the diameter AB at point Q, then ∠APO=∠OPQ. As point P approaches point B infinitely, where does point Q approach?'

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'Prove that among quadrilaterals inscribed in a circle, the one with the largest area is a square.'

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"Regarding inversion with respect to circle O, let's assume that point P moves to point P' through this inversion (we call this circle O the inversion circle). At this point, points P and P' are reflections of each other with respect to circle O, or we can also say that point P' is the reflection of point P. As shown in the diagram on the right, through inversion with respect to circle O, points inside circle O move to points outside the circle, and points outside the circle move to points inside the circle. Additionally, points on circle O remain unchanged through inversion. It is important to note that according to the definition of inversion mentioned above, the destination of the center O cannot be determined, but sometimes we consider the destination of the center O as a virtual point at infinity, a concept that is studied in university-level mathematics. Drawing inversions Through inversion with respect to circle O, the point P moving to point P' can be drawn by following these steps. Understanding this drawing process makes it easier to grasp the concept of inversion. [1] Let A be the intersection point of the line passing through point P and perpendicular to OP with circle O. [2] Let P' be the intersection point of the tangent to circle O at point A and the line OP. From △OAP' ~ △OPA, we have OA:OP = OP':OA, hence OP ⋅ OP' = OA² = r² (radius of the circle squared). Note that even when point P is outside the circle, point P' can still be drawn using the same method. Properties related to the inverse of circles and lines Inversion with respect to circle O has the following four properties: (1) Circles passing through the center O of the inversion circle are mapped to lines not passing through O. (2) Lines not passing through the center O of the inversion circle are mapped to circles passing through O. (3) Circles not passing through the center O of the inversion circle are mapped to circles not passing through O. (4) Lines passing through the center O of the inversion circle are mapped to the lines themselves."

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'Investigate how to express the position vectors of the five centers (centroid, incenter, orthocenter, circumcenter, and excenter) of a triangle with vertices at points A(𝐚), B(𝐛), C(𝐜) using 𝐚, 𝐛, and 𝐜.'

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'Important Topic 40 | Comparison of Vector Magnitudes'

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'Specify the condition for the line segment AB to be perpendicular to CD.'

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'Study the parameter representation of an ellipse, express it with only x and y by eliminating t.'

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'Let quadrilateral ABPC inscribed in a circle satisfy the following conditions (a), (b):\n(a) Triangle ABC is an equilateral triangle.\n(b) The intersection of AP and BC divides line segment BC into p:(1-p) [0<p<1].\nExpress vector AP in terms of vectors AB, AC, and p.'

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'Explain about the mathematicians who used ancient Greek methods for calculating areas and volumes.'

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'Important Example 57 | Equation of a plane\nFind the equation of a plane passing through points A(0,1,-1), B(4,-1,-1), and C(3,2,1).'

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'Prove that the hexagon is inscribed in a conic section.'

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'When a point z moves along the circle with radius 1 centered at the origin O, what kind of figure does the point w represented by w=(1-i) z-2 i draw?'

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'Find the equation of the plane passing through the following three points:\n57 (1) A(1,0,2), B(0,1,0), C(2,1,-3)\n(2) A(2,0,0), B(0,3,0), C(0,0,1)'

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'For |x - π/2|, when the enclosed part is the gray area in the right figure and is symmetric with respect to the line x = π/2, find the volume V.'

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'Let ABCD be a quadrilateral in the plane. If the diagonals AC and BD are perpendicular, prove the following:\n(1) Let $\\overrightarrow{AB}=\\vec{b}$, $\\overrightarrow{AC}=\\vec{c}$, $\\overrightarrow{AD}=\\vec{d}$, then $\\vec{b} \\cdot \\vec{c}-\\vec{c} \\cdot \\vec{d}=0$.\n(2) $AB^{2}+CD^{2}=AD^{2}+BC^{2}$.'

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'Let ABCD be a quadrilateral with diagonals AC and BD, and a circle with center O circumscribed around quadrilateral ABCD. Let vectors OA, OB, OC, OD be denoted by a, b, c, d.'

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'Prove that if the line segment $PO$ and $QO$ passing through the ends of the chord $PQ$ on the parabola $y^{2}=4 p x(p>0)$ and the origin $O$ are orthogonal, then the chord $PQ$ passes through a fixed point.'

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'If the curve $f(x, y)=0$ satisfies $f(x,-y)=f(x, y)$, it is symmetric about the x-axis, and if $f(-x, y)=f(x, y)$, it is symmetric about the y-axis. \nLet the coordinates of point Q on the curve be $(x, y)$ and derive the relationship between x and y.'

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'Chapter 3 Reflection on Geometry and Equations Method for finding the tangent of a circle'

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'Given a line segment AB of length 4. Find the locus of point P as it moves while satisfying the equation 2AP² - BP² = 17 with points A, B and moving by 71.'

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'Given that the length of the perpendicular dropped from point (1,1) to the line ax - 2y - 1 = 0 is √2, find the value of the constant a.'

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'Find the acute angle θ formed by the two lines x+3y-6=0 and x-2y+2=0.'

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'Reflection symmetry, distance between a point and a line'

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'Practice (4) 127 For the line y = a x+1-a^2/4. (1), when a varies over all real values, illustrate the region through which the line (1) can pass.'

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'Find the equation of the inscribed circle of the triangle formed by the lines x=3, y=2, and 3x-4y+11=0.'

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'On the xy-plane, let three distinct points P1(a1, b1), P2(a2, b2), P3(a3, b3) be taken, excluding the origin. Furthermore, consider three lines l1: a1x+b1y=1, l2: a2x+b2y=1, l3: a3x+b3y=1.'

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"Let's think about how to prove the addition theorem and double angle formulas using geometric figures. Although the range of α, β, and θ is limited, it is interesting to see the geometric meaning of the addition theorem."

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'Plot the locus of the point (x+y, x-y) as the real numbers x, y change satisfying the following conditions: (1) -1 ≤ x ≤ 0, -1 ≤ y ≤ 1'

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'For the triangle ABC with vertices A(6,13), B(1,2), C(9,10): (1) Find the equation of the line passing through point A that divides the area of triangle ABC into two equal parts. (2) Find the equation of the line passing through a point P that divides the side BC internally in 1:3 ratio and divides the area of triangle ABC into two equal parts.'

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"Substitute x=3 into the equation 3x-4y+11=0 to get y=5, and substitute y=2 into the equation 3x-4y+11=0 to get x=-1. Therefore, the coordinates of the vertices of the triangle are (-1,2),(3,2),(3,5). Let r be the radius of the circle to be found, and the coordinates of the center are represented as (3-r,r+2), satisfying -1<3-r<3 and 2<r+2<5, which is solved to be 0<r<3. The distance between the line 3x-4y+11=0 and the center of the circle is equal to the radius of the circle, which gives the equation |3(3-r)-4(r+2)+11|/√(3^2+(-4)^2)=r. Solving this, we get |12-7r|=5r, then 12-7r=±5r, leading to r=1. When r=1, the center's coordinates are (2,3), and the equation of the circle is (x-2)^2+(y-3)^2=1"

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'Consider the solution for finding the equation of a circle problem.'

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'Take the point A(-3,0) and consider two points B and C that satisfy the following conditions for 0°<θ<120°.'

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'Prove that the three medians of triangle ABC intersect at one point. Show that in triangle ABC, 2AB^2 < (2 + AC^2)(2 + BC^2) holds true.'

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'There is a line segment AB with a length of 4. Point P moves in such a way that the equation 2AP^2 - BP^2 = 17 is satisfied for points A and B. Find the locus of point P.'

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'(1) Find the acute angle \ \\theta \ formed by the two lines \ x+3 y-6=0, x-2 y+2=0 \. \n(2) The line \ y=-x+1 \ makes an angle of \ \\frac{\\pi}{3} \ and passes through the point \\( (1, \\sqrt{3}) \\). Find the equation of this line.'

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'The coordinates of the intersection are found by solving the system of equations between the line perpendicular to 4x + 3y = 5√2 passing through (0,0) and the line 3x - 4y = 0.'

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'Find the locus of points for which the angle APB passing through points A, B is a constant value α.'

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'Prove that the perpendiculars dropped from each of the three vertices of triangle ABC to its opposite side or its extension intersect at one point (this point of intersection of the three perpendiculars is called the orthocenter of the triangle).'

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"What is a CHART? In C.O.D. (The Concise Oxford Dictionary), CHART is described as Navigator's sea map, with coast outlines, rocks, shoals, etc. Based on the chart editing policy shown below, find the key points of the problem and explain a way to come up with a solution."

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'Proof Using Coordinates (2) Prove that the perpendicular bisectors of each side of triangle ABC intersect at a single point.'

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'(1) Prove that the three medians of triangle ABC intersect at one point. (2) In triangle ABC, prove that 2AB²<(2+AC²)(2+BC²) holds true.'

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'Prove the following inequality regarding the value of the mathematical constant π. Do not use π=3.14…… . [University of Oita]'

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'Show that the curve C is a hyperbola by finding the equation of the curve obtained by rotating C: x^2 + 6xy + y^2 = 4 around the origin by π/4.'

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'By finding the equation of the curve obtained by rotating the origin as the center by π/4, prove that the curve C is a hyperbola.'

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'Prove that in quadrilateral OABC, if OA squared plus BC squared equals OC squared plus BA squared, then OB is perpendicular to AC.'

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'Prove that in triangle ABC, when the midpoints of sides AB and AC are D and E respectively, BC is parallel to DE and BC=2DE (Midpoint Theorem).'

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'Find the equation of the curve passing through the point (1,1) where the tangent at point P on the curve passing through (1,1) intersects the x-axis and y-axis at points Q, R respectively, with O as the origin. Given that the curve is in the first quadrant and always satisfies △ORP = 2△OPQ.'

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'Prove that the sum of distances OA + OB from the origin O to the points A, B where the tangents drawn to the point P (not on the coordinate axes) on the curve \\sqrt{x} + \\sqrt{y} = \\sqrt{a} (a > 0) intersect the x-axis and y-axis respectively, remains constant.'

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'Prove that in quadrilateral OABC, if OA^2 + BC^2 = OC^2 + BA^2, then OB is perpendicular to AC.'

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'Find the locus of points that satisfy the following condition: the distance from point F is equal to the distance from the line l. Here, F is at (c, 0), and l is the y-axis (x=0).'

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'Draw a chord AB parallel to the minor axis passing through the foci of the ellipse. Prove that the square of the length of the minor axis is equal to the product of the length of the major axis and the chord AB.'

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'Prove that when three distinct points A(α), B(β), C(γ) on the unit circle and a point H(z) not on the circle satisfy the equation z=α+β+γ, then H is the orthocenter of △ABC.'

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'(1) Let the curve C be represented by the parametric equations x=2(t+1/t+1) and y=t-1/t. Find the equation of the curve C and sketch its general shape. [The University of Tsukuba]'

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'Prove that the sum of 1/FP and 1/FQ passing through one of the foci F of a quadratic curve is constant regardless of the direction of the chord.'

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'Show the formula for finding the angle of rotation from ray AB to ray AC with respect to three different points A(alpha), B(beta), and C(gamma).'

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'Given A(0), B(β), C(γ), D(β/2), E(γ/2), and (γ/2 - β/2) / (γ - β) = 1/2, we have BC parallel to DE and DE = |γ/2 - β/2| = 1/2|γ - β|. Therefore, BC = 2DE.'

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'If the line segments PO and QO passing through the two ends of the chord PQ of the parabola y^2 = 4px (p > 0) and the vertex O are perpendicular, then prove that chord PQ passes through a fixed point.'

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'Draw a chord AB parallel to the minor axis passing through the foci of an ellipse. Prove that the square of the minor axis length is equal to the product of the major axis length and the length of chord AB which is 49.'

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"＜Ptolemy's theorem＞For a quadrilateral ABCD inscribed in a circle, the following equation holds:\n\\n\\mathrm{AB} \\cdot \\mathrm{CD}+\\mathrm{AD} \\cdot \\mathrm{BC}=\\mathrm{AC} \\cdot \\mathrm{BD}\n\"

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'There are two moving points P, Q on the circumference of a circle with radius a. P, Q simultaneously depart from a fixed point A on the circumference and move in opposite directions to the clock. Let O be the center of the circle. When the ratio of the angular velocities of the radii OP and OQ is constant at 1:k (k > 0, k ≠ 1), find the polar equation of the trajectory of the midpoint M of segment PQ. When P and Q coincide, point M represents point P (Q).'

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'Take the point Pn(xn, yn) (n=0,1,2, …) on the two-dimensional coordinate plane satisfying the following conditions (A), (B). 90(A) (x0, y0)=(0,0), (x1, y1)=(1,0) (B) For n ≥ 1, the vector PnPn+1 has a length that is half of the vector Pn-1Pn and points in the direction obtained by rotating Pn-1Pn counterclockwise by 90 degrees. In this case, the limits of xn and yn as n approaches infinity are lim{n→∞}xn= A, lim{n→∞}yn= B. [Meiji University]'

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'Let A(α) and B(β) be two distinct points. When m>0, n>0, and m≠n, the set of all points P(z) satisfying the equation n|z-α|=m|z-β| consists of points that internally or externally divide the line segment AB in the ratio m:n, with these points as the endpoints of the diameter of a circle (Apollonius circle). Prove this statement.'

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'Find the locus of point $Q$ when two tangents drawn from an external point $C_1$ intersect perpendicularly.'

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'Prove that the triangle OAB with vertices O(0), A(1), and B(i), where angle O is a right angle, and the triangle PQR with vertices P(α), Q(β), R(γ), where angle P is a right angle, satisfy the equation 2α²+β²+γ²-2αβ-2αγ=0.'

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'Let a be a constant greater than 1. Find the area S of the region enclosed by the curve x^2-y^2=2 and the line x=\\sqrt{2} a by considering a rotation of \\frac{\\pi}{4} centered at the origin.'

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'In triangle ABC, let M be the midpoint of side BC. The following equation holds.'

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'Consider the line $\\ell: \\frac{x}{a}+\\frac{y}{b}=1$ and the curve $C: \\sqrt[3]{\\frac{x}{a}}+\\sqrt[3]{\\frac{y}{b}}=1$, where $a$ and $b$ are positive constants.'

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"There is a point A and a line g that does not pass through A on the plane. Let's denote the curve formed when a point P moves on the plane such that the distance from P to g is equal to the length of the line segment PA as K. Let G and H be the points where two tangents can be drawn from point Q on curve K. Prove the following: (1) If point Q is on the line g, then the line GH passes through point A. (2) If the line GH passes through point A, then point Q lies on the line g. [From Oita Medical University]"

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'Furthermore, point R lies on the circle with diameter PQ, so ∠PRQ = π/2'

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"How to find the position vector of the intersection using Menelaus's theorem and Ceva's theorem?"

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"In an ellipse with major axis length 2a, center O, and minor axis BB', let P be a point on the ellipse other than B and B'. Let BP and B'P intersect the major axis or its extension at points Q and R, respectively, so that OQ・OR=a². Prove this using circles."

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'Find the equation of the plane passing through the three points A(0,1,-1), B(4,-1,-1), and C(3,2,1).'

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'Find the equation of the plane passing through the origin O and perpendicular to the z-axis.'

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'On the plane, consider the curve \ C \ given by the polar equation \ r=\\frac{1}{1+\\cos \\theta} \.'

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'Prove that the tangent at a point $\\mathrm{P}(x_{1}, y_{1})$ on the hyperbola $\\frac{x^{2}}{16}-\\frac{y^{2}}{9}=1$ bisects the angle $\\angle \\mathrm{FPF}^{\\prime}$ formed by the point $\\mathrm{P}$ and the two foci $\\mathrm{F}, \\mathrm{F}^{\\prime}$. Here, $x_{1}>0, y_{1}>0$.'

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'(1) Point z is the perpendicular bisector of the line segment connecting points 1 and i. (2) Point z is a circle with center at point 1-i and radius 2.'

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"Using the complex plane, prove the following theorem. For a quadrilateral ABCD inscribed in a circle with a radius of 100 units, the equation AB·CD+AD·BC=AC·BD holds (Ptolemy's theorem)."

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'Find the equation of a plane passing through point C(0,3,-2) and perpendicular to the z-axis.'

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'Example 46 | Coplanar Condition Determine the value of x so that the following 4 points are coplanar: A(1,3,3), B(1,1,2), C(2,3,2), P(x, x, x) [Similar to Keio University] The condition for point P to be on the plane of ABC for three non-collinear points A, B, C is that one of the following conditions holds: In the case where the origin is O, [1] there exist real numbers s, t such that \\\overrightarrow{AP}=s\\overrightarrow{AB}+t\\overrightarrow{AC}\ [2] there exist real numbers s, t, u such that \\\overrightarrow{OP}=s\\overrightarrow{OA}+t\\overrightarrow{OB}+u\\overrightarrow{OC}, s+t+u=1\ Represent [1] or [2] with components, and reduce to an equation problem.'

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'Let \ \\mathrm{O} \ be the origin. There is a fixed point \\( \\mathrm{A}(k, 0)(k>0) \\) on the \ x \ axis. Now, consider a moving point \ \\mathrm{P} \ on the plane such that \\( \\overrightarrow{\\mathrm{OP}} \\neq \\overrightarrow{0}, \\overrightarrow{\\mathrm{OP}} \\cdot(\\overrightarrow{\\mathrm{OA}}-\\overrightarrow{\\mathrm{OP}})=0,0^{\\circ} \\leqq \\angle \\mathrm{POA}<90^{\\circ} \\) . Find (1) the equation representing the trajectory of point \\( \\mathrm{P}(x, y) \\) using \ x, y \. (2) Determine the maximum value of \ |\\overrightarrow{\\mathrm{OP}}||\\overrightarrow{\\mathrm{OA}}-\\overrightarrow{\\mathrm{OP}}| \ and the corresponding value of \ \\angle \\mathrm{POA} \. [Saitama Institute of Technology]'

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'From a height of 30m above the water surface on a vertical wall on the waterfront, a boat is pulled with a rope of length 58m. Calculate the speed of the boat 2 seconds after turning the rope at 4m/s.'

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'In the Cartesian coordinate plane, the region represented by the simultaneous inequalities x ≥ 0, y ≥ 0, x + 3y ≤ 3, 1 ≤ 2x + y ≤ 2 is the shaded area in the diagram below, including the boundary lines.'

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'Prove that when three distinct points A(α), B(β), and C(γ) on the unit circle and a point H(z) not on the circle satisfy the equation z=α+β+γ, then H is the orthocenter of △ABC.'

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'Prove the orthocenter of a triangle\nGiven three distinct points A(α), B(β), C(γ) on the unit circle and a point H(z) not on the circle, when the equation z = α + β + γ holds, prove that H is the orthocenter of triangle ABC.'

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'Find the equation of the curve obtained by moving the hyperbola $x^{2}-2 y^{2}=-4$ symmetrically about the point (-3,1).'

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'Take a point O on the line segment AB (excluding the two ends), construct squares AOCD and OBEF with sides AO, OB on the same side of the line segment AB. Prove that AF⊥BC using the complex number plane.'

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'Prove that the area of triangle $\\triangle \\mathrm{OQR}$ formed by the intersection points $\\mathrm{Q}, \\mathrm{R}$ of the tangent line at point $\\mathrm{P}(x_{0}, y_{0})$ on the hyperbola $b^{2} x^{2}-a^{2} y^{2}=a^{2} b^{2}(a>0, b>0)$ and the asymptote, with the origin as $\\mathrm{O},$ is independent of the choice of point $\\mathrm{P}.$'

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'Equation of a plane\n(1) The equation of a plane passing through a point \\( \\mathrm{A}\\left(x_{1}, y_{1}, z_{1}\\right) \\) and perpendicular to a non-zero vector \\( \\vec{n}=(a, b, c) \\) is \\( a\\left(x-x_{1}\\right)+b\\left(y-y_{1}\\right)+c\\left(z-z_{1}\\right)=0 \\)\n(2) General form is \ a x+b y+c z+d=0 \ where \\( (a, b, c) \\neq(0,0,0) \\)'

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'(2) Find the equation of a hyperbola with asymptotes of the two lines $y=\\frac{\\sqrt{7}}{3} x, y=-\\frac{\\sqrt{7}}{3} x$ and foci at points $(0,4),(0,-4)$.'

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'For a positive number a, consider the tangent line to the parabola y=x^{2} at point A(a, a^{2}), which is obtained by rotating point A by -30°. Let line l be this rotated line. Let B be the point of intersection of line l and the parabola y=x^{2} which is not A. Furthermore, let C(a, 0) and O be the origin. Find the equation of line l. Also, let S(a) be the area enclosed by line segments OC, CA, and the parabola y=x^{2}, and let T(a) be the area enclosed by line segment AB and the parabola y=x^{2}. Find c=lim_{a→∞} (T(a)/S(a)).'

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'Practice problem: When the ellipse (x²/a²) + (y²/b²) = 1 (a > b > 0) and the line y = mx + n have exactly one common point, then n² = a²m² + b². Using a circle, prove this.'

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'Given quadrilateral ABCD with diagonals AC and BD, and a circle with center O circumscribed around quadrilateral ABCD. Let vectors OA, OB, OC, OD be denoted by a, b, c, d.\n(1) If the magnitudes of vectors a+b+c and a+b+d are equal, prove that sides AB and CD are parallel or point O lies on side AB.\n(2) If the centroids of triangles ABC, BCD, CDA, DAB are equidistant from point O, prove that quadrilateral ABCD is a rectangle.'

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'(1) Find the equation of the line passing through point A(1,2) and perpendicular to BC, where A(1,2), B(2,3), and C(-1,2).\n(2) Find the acute angle α formed by the two lines x-2y+3=0 and 6x-2y-5=0.'

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'Consider a circle with center O. On the circumference of this circle, there are 3 points A, B, C such that OA+OB+OC=0. The task is to prove that triangle ABC is an equilateral triangle. Let the radius of the circle be r (r>0), then |OA|=|OB|=|OC|=r. From OA+OB+OC=0, we have OA+OB=-OC. Therefore, |OA+OB|^{2}=|-OC|^{2}, which simplifies to |OA|^{2}+2OA·OB+|OB|^{2}=|OC|^{2}. Hence, r^{2}+2OA·OB+r^{2}=r^{2}, which leads to OA·OB=-r^{2}/2. In this case, |AB|^{2}=|OB-OA|^{2}=|OB|^{2}-2OA·OB+|OA|^{2}=r^{2}-2(-r^{2}/2)+r^{2}=3r^{2}. Since |AB|>0, |AB|=sqrt{3}r. Similarly, |BC|=|CA|=sqrt{3}r. Therefore, AB=BC=CA. Thus, triangle ABC is an equilateral triangle.'

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'Express the following equations in polar form: (1) x+y+2=0 (2) x²+y²-4 y=0 (3) x²-y²=-4'

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'In a regular hexagon ABCDEF, let M be the midpoint of side DE. Let O be the intersection point of the diagonals AD, BE, and CF. The following vector relationships hold true:'

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'The distance between point A(x₁, y₁) and the line ax + by + c = 0 is |ax₁ + by₁ + c| / √(a² + b²). The vector 𝑛 perpendicular to the line ax + by + c = 0 is 𝑛 = (a, b). p.343 Basic concepts 1.'

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'Find the equation of the curve obtained by enlarging the original circle by a factor of 5/2 in the x-direction with respect to the y-axis.'

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'Prove that the midpoints of the sides AB, BC, CD, DA of quadrilateral ABCD denoted by P, Q, R, S respectively, and the midpoints of the diagonals AC, BD denoted by T, U coincide for line segments PR, QS, TU.'

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'Let △ABC be in the plane. Let O be the circumcenter of △ABC with circumradius R. Also, let point H be a point that satisfies ∽OA+∽OB+∽OC=∽OH, where point H is different from points A, B, and C. Let ∽OA=𝐚, ∽OB=𝐛, ∽OC=𝑐. (1) Express ∽AH and ∽CH in terms of 𝐚, 𝐛, 𝐜, and show that ∽AH is perpendicular to BC and ∽CH is perpendicular to AB. (2) Let the midpoint of segment OH be denoted as P, and the midpoints of the sides AB, BC, CA of △ABC be denoted as L, M, N, respectively. Express ∽PL, ∽PM, ∽PN in terms of 𝐚, 𝐛, 𝐜, and show that P is the circumcenter of △LMN. (3) Let D be the midpoint of segment AH, and show that P is the midpoint of segment DM. (4) Draw a perpendicular line from vertex A to line BC, let the intersection point be E. Show that point E lies on the circumference of the circumcircle of △LMN.'

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'Prove that 1/PF*QF + 1/RF*SF is constant when two chords PQ and RS passing through one of the foci F of a conic intersect at right angles.'

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'Find the equation of the curve which is formed by enlarging the circle x^2+y^2=4 by a factor of 5/2 along the x-axis with y-axis as the reference.'

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'(1) For the points A(-1,4), B(-4,-3), and C(8,3), find the equation of the line passing through point A and perpendicular to BC. (2) Find the acute angle α formed by the lines l1: x-√3y+3=0 and l2: √3x+3y+1=0.'

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'Prove the following geometric properties:\n1) Prove that vertical angles are congruent.\n2) When line $\\ell$ and $m$ intersect line $n$, if $\\ell \\parallel m$, then the corresponding angles and alternate interior angles are congruent.\n3) Regarding lines $\\ell$ and $m$, if one pair of corresponding angles or alternate interior angles are congruent, then prove that $\\ell \\parallel m$.'

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'When taken on the edge x, the coordinates of each vertex can be represented as A(a, b), B(-c, 0), C(c, 0).\n(1)\n\\[\n\egin{aligned}\n\\mathrm{AB}^{2}+\\mathrm{AC}^{2} & =\\left\\{(a+c)^{2}+b^{2}\\right\\}+\\left\\{(a-c)^{2}+b^{2}\\right\\} \\\\\n& =2 a^{2}+2 c^{2}+2 b^{2} \\\\\n& =2\\left(a^{2}+b^{2}+c^{2}\\right) \\\\\n2\\left(\\mathrm{AM}^{2}+\\mathrm{BM}^{2}\\right) & =2\\left\\{\\left(a^{2}+b^{2}\\right)+c^{2}\\right\\} \\\\\n& =2\\left(a^{2}+b^{2}+c^{2}\\right)\n\\end{aligned}\n\\]\nTherefore, \\( \\quad \\mathrm{AB}^{2}+\\mathrm{AC}^{2}=2\\left(\\mathrm{AM}^{2}+\\mathrm{BM}^{2}\\right) \\)'

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'(3) In Figure 2, determine the total number of different paths from point A to point B. It should be noted that diagonal paths are not allowed.'

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'In the diagram on the right, a square is divided into 5 regions by connecting the midpoints of each side, totaling 14 smaller regions. When adjacent regions are colored with different colors, how many different ways are there to color in the following scenarios? It is assumed that coloring schemes that are identical after rotation are considered the same. (1) Select 2 colors out of 4 different colors to color. (2) Select 3 colors out of 4 different colors and use all 3 colors for coloring.'

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'Practice 43: From page 339 of this book, regarding the area, since △ABP=¼△ABC, it follows that BP:BC=1:3. Therefore, BP:PC=1:2. Moreover, given AB:AC=1:2 from the conditions. Hence, AP is the angle bisector of ∠BAC. Thus, ∠BAP=½∠BAC=½ × 90°=45°.'

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'Prove that if the intersection of the two bisectors of the two vertices ∠A and ∠C of quadrilateral ABCD lies on the diagonal BD, then the intersection of the two bisectors of the two vertices ∠B and ∠D lies on the diagonal AC.'

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'Example 48 Five centers relationship In an acute-angled triangle ABC, the circumcenter is O, the orthocenter is H, and the centroid is G. (1) Let OM and ON be the perpendiculars from O to sides AB and BC, respectively. Let L be the midpoint of segment BH. Prove the following: (a) The quadrilateral MLNO is a parallelogram. (b) AH = 2ON (2) If triangle ABC is not an equilateral triangle, prove that the points G, O, H are collinear, and G divides segment OH internally in the ratio 1:2.'

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'What are the possible position relationships between a line ℓ and a plane α?'

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"Prove that in the figure on the right, with point P as the point of tangency, the intersection points of the two circles O and O' with the common external tangent are C and D. Let A and B be the intersection points of the line passing through point P with the two circles O and O' other than P, then AC ⊥ BD."

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'Exercise 19 ⇒ This book p.282 Let one side of a cube be defined as one grid, move one grid to the right, back, and up respectively represented by →, ↗, ↑. (1) As shown in the right figure, assuming there is a path in the upper left corner, the path from A to B is a permutation of → 3 and ↗ 2, so it is 5!/(3!2!)=10. Among them, the path passing through the × mark in the figure is 1, so the number of paths to find is 10-1=9 (ways). (2) Be careful not to count the cases where both point C and point D pass through twice. (Total)-(Pass point C or point D) Can be solved in the same way (1) and (2). Consider the paths to B passing through the diagonal part of Figure 2. It is possible to create a virtual path or use other methods like an alternative solution.'

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'Utilization of a quadrilateral inscribed in a circle of radius 36'

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'The minimum value of AC+BC occurs when points A, B, C are collinear in the unfolded diagram of a regular octahedron depicted in the right figure [3]. In this case, ∠ACB is equal to ∠PRQ in figure [2]. Therefore, in triangle PQR, according to the cosine rule'

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'Prove that FG = FE when drawing the tangent FG from point F to the circle.'

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'In triangle ABC, let the intersection of the angle bisector of ∠B and side AC be named as D, and the intersection of the angle bisector of ∠C and side AB be named as E.'

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"Area of a quadrilateral inscribed in a circle\nWhen finding the area of a quadrilateral inscribed in a circle, there is a formula similar to Heron's formula. Let's consider this formula based on the recent exam question.\nLet the quadrilateral ABCD be inscribed in a circle. Denote the lengths of sides DA, AB, BC, and CD as a, b, c, d, respectively, and let ∠DAB=θ. Also, let T be the area of the quadrilateral ABCD.\n(1) Prove that a²+b²-c²-d²=2(ab+cd)cosθ.\n(2) Prove that T=√[(s-a)(s-b)(s-c)(s-d)]. Where s=(a+b+c+d)/2.\n[University of Yamaguchi]"

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'[2] When $AP = PQ$, if we consider moving point $P$ clockwise from point $A$, there are 5 ways to choose point $P$ such that $AP = PQ$.'

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'In △ABC, if the midpoint of side BC is M, then the following equation holds: \n\nAB^{2}+AC^{2}=2(AM^{2}+BM^{2}) (Median Theorem)'

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'From the chord intersection theorem and BC=BD, we have ∠CBT=∠BDC=∠BCD=68°. Also, ∠DBC=180°-(68°+68°)=44°, ∠ACB=∠ADB=115°-68°=47°. Therefore, θ=180°-(44°+47°)=89°.'

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'Given two parallel lines ℓ and m, a line n intersecting them, and two points A and B. Let point C be on line ℓ and point D be on line m, so that CD is parallel to n and AC is perpendicular to BD. Construct points C and D.'

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'To understand the relationship between the sets of siblings intuitively, it is best to draw a diagram. Let the universal set be U, and the sets of brothers, sisters, younger brothers, and younger sisters be P, Q, R, and S respectively.'

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'(3) Prove the congruence of triangle DAE and triangle CAB.'

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'(3) Let the intersection of line AH and side BC be D, and the intersection of line CH and side AB be E, then'

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'In triangle ABC, let D be the intersection point of the angle bisector of angle A and side BC, and let E and F be the intersection points of the angle bisectors of angles ADB and ADC with sides AB and AC, respectively. Prove the following: (1) Triangle BEF: Triangle AEF = BD: AD (2) Triangle BEF: Triangle CEF = AB: AC'

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'On the sides AB, BC, CA of triangle ABC, points D, E, F are taken such that AD:DB=t:(1-t), BE:EC=2t:(1-2t), CF:FA=3t:(1-3t), and the area of triangle DEF is denoted as S. Answer the following questions.'

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'Proof of important example 101 Equality of sides and angles\nIn triangle ABC, prove that the following equation holds:\n\\[\n(a-b \\cos C) \\sin A=(c-b \\cos A) \\sin C\n\\]\nExamples 62, 63\nAs mentioned in guidance D.172 and example 55, to prove the equation P=Q, the following methods can be used:\n[1] Transform either P or Q to derive the other.\n[2] Transform both P and Q to derive the same expression.\n[3] Transform P-Q to show that it equals 0.\nThe choice of method depends on the given problem, but here we demonstrate the approach of [3]. Thus, in order to simplify P-Q, we consider reducing the characters, which involves\nEliminating angles and converting the problem to one involving only sides\nFor this, we can use formulas such as \ \\sin A=\\frac{a}{2 R}, \\cos A=\\frac{b^{2}+c^{2}-a^{2}}{2 b c} \.'

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'The following (1), (2), (3) are known as the three great geometric problems of Greece. (1) Trisection problem of an angle: To divide a given angle into three equal parts (2) Doubling the cube problem: To construct a cube with twice the volume of the given cube (3) Squaring the circle problem: To construct a square with the same area as the given circle.'

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'(2) Since ∠ABI = ∠IBC and ∠ACI = ∠ICB, in △IBC, α = 180° - (1/2 ∠B + 1/2 ∠C) = 180° - 1/2(∠B + ∠C) = 180° - 1/2(180° - ∠A) = 180° - 90° + 42°/2 = 111°'

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'Prove that the line connecting the intersection point O of BE and CD passes through the midpoint of side BC in triangle ABC, where line BC is parallel to line AB and AC intersect at points D and E.'

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'Prove that when quadrilateral ABCD is inscribed in a circle, there exists a point E on diagonal BD such that ∠BAE=∠CAD.'

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'Practice (1) Prove that when an arbitrary point O is taken inside ΔABC, and the angle bisectors of ∠BOC, ∠COA, ∠AOB intersect the sides BC, CA, AB at points P, Q, R respectively, then AP, BQ, CR intersect at a single point.\n(2) Let D be the intersection point when the external angle bisector of ∠A in ΔABC extends side BC. Let E, F be the intersections of the angle bisectors of ∠B, ∠C with sides AC, AB respectively. Show that the three points D, E, F are collinear.'

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'Prove that line l is perpendicular to plane OAB when line OA is perpendicular to plane alpha and line l lies on plane alpha.'

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'Therefore, by the power of a point theorem, AB²=AD×AE'

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'Find the coordinates of the point symmetrical to (2, 3) with respect to the line x = 1 and the new equation.'

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'Please explain how to draw 6 line segments between parallel lines.'

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'Please explain the Venn diagrams of 4 sets from the exercises in Chapter 2 on sets and propositions.'

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'Proof: If the line l is perpendicular to the two intersecting lines m and n on plane α, then line l is perpendicular to plane α.'

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'Proof of the ratio of the bisector of angle 42 and the segment'

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'In quadrilateral ABCD, let P be the intersection point of AC and BD. Given that ∠APB=∠CPD=90° and AB//DC. Prove that the circumcircles of △PAB and △PCD are mutually tangent.'

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'Example 53 | Shortest path of broken line\nA point P is inside the acute angle ∠XOY. Points Q and R are taken on the half-lines OX and OY (excluding O) respectively. How to position Q and R to minimize PQ+QR+RP?'

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'Please solve the following problems:\n(1) Using the cosine rule, find the length of BD.\n Given values: BC = 4, CD = 3√2, ∠BCD = 45°\n\n(2) Find the area of quadrilateral ABCD. Given that quadrilateral ABCD is inscribed in a circle and ∠BAD = 135°.'

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"The five centers of a triangle and the inverse of Ceva's theorem"

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'By translation, this point will move to point (2+2, -4-1) which is point (4, -5), so the required equation of the parabola is y=(x-4)^2-5 or y=x^2-8x+11.'

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'In triangle ABC, let D be the intersection point of the angle bisector of angle B and side AC, and let E be the intersection point of the angle bisector of angle C and side AB. Prove that if angle B is less than angle C, then BD > CE.'

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'Therefore, the diagonals AF, DE of the quadrilateral ADFE intersect perpendicularly and bisect each other. Thus, the quadrilateral ADFE is a rhombus.'

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'Circumferential angle, square inscribed in a circle'

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'In Figure 1, how many different ways are there to go from point A to point B via the shortest path without passing through either point C or point D?'

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'Prove that a line perpendicular to one of two parallel lines is also perpendicular to the other.'

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'Example 41 Translation and Symmetric Movement\nMove a parabola by symmetry about the origin, then parallel move -2 in the x-axis direction, and 3 in the y-axis direction, the parabola will move to y=3x^2-6x+5. Find the equation of the original parabola.'

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De Morgan's laws
For the subsets A and B of the universal set U
\overline{A \cap B}=\overline{A} \cup \overline{B}, \quad \overline{A \cup B}=\overline{A} \cap \overline{B} \quad (De Morgan's laws)
hold. Verify this using a diagram.

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Find the polar equations of the following lines.
(1) The line passing through point A(3/2, 0) on the initial line OX and perpendicular to the initial line.
(2) The line passing through the pole O and making an angle of -π/4 with the initial line.

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Given three distinct points \( \mathrm{A}(lpha), \mathrm{B}(eta), \mathrm{C}(\gamma) \) on the unit circle and a point \( \mathrm{H}(z) \) not on the circle, prove that $\mathrm{H}$ is the orthocenter of $riangle \mathrm{ABC}$ when the equation z=lpha+eta+\gamma holds.

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We covered vector equations in the plane on pages 55 and 56. Here, let's consider plane equations and vector equations in space.
1 Plane equation
As mentioned in the reference on page 78, a plane passing through three points not on the same straight line is uniquely determined. This can also be defined by using a point A and a non-zero vector n. There are innumerable lines passing through point A and perpendicular to vector n. These innumerable lines make up the plane. Let's derive the plane equation.
Let P(x, y, z) be a point on the plane that passes through A(x1, y1, z1) and is perpendicular to the non-zero vector n(a, b, c).
(1) When A and P do not coincide, from n⊥AP, we have n·AP=0. Since AP=(x-x1, y-y1, z-z1), it follows that n·AP=0 gives
a(x-x1)+b(y-y1)+c(z-z1)=0 (*
When A and P coincide, from AP=0, we have n·AP=0, therefore (*) holds.
(*) is the plane equation passing through point A and perpendicular to vector n. Vector n is called the normal vector to the plane.
(2) Rearranging (*) from (1)
a*x + b*y + c*z - a*x1 - b*y1 - c*z1 = 0
Let -a*x1 - b*y1 - c*z1 = d
a*x + b*y + c*z + d = 0 \longleftarrow -a*x1 - b*y1 - c*z1 is a constant.
This is sometimes called the general form of the plane equation.

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[Proof of the position vector of the external division point] Assume the case where $m > n$. The case where $m < n$ is similar. Let the point \( \mathrm{P}(ec{p}) \) divide the segment $\mathrm{AB}$ externally in the ratio $m$. Since \( \mathrm{AP}: \mathrm{AB} = m:(m-n) \), we have \overrightarrow{\mathrm{AP}} = rac{m}{m-n} \overrightarrow{\mathrm{AB}} . Hence, \( ec{p} - ec{a} = rac{m}{m-n} (ec{b} - ec{a}) \). The formula for the external division point is obtained by substituting $n$ with $-n$ in the formula for the internal division point.

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In the xy-plane, when the ellipse x²/4 + y² = 1 is translated 1 unit in the x-direction and a units in the y-direction, and the resulting ellipse passes through the origin, a=□.

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Find the acute angle formed by the lines x-√3y+3=0 and √3x+3y+1=0.

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From any point P on a hyperbola, draw perpendicular lines PQ and PR to the two asymptotes. Prove that the product of the lengths of these line segments PQ · PR is a constant.

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Place a point P on the line segment BF, given its y-coordinate as a. Let H be the intersection of the perpendicular dropped from P to the line CE, and the perpendicular dropped from point C to the line EP. Express the vector EP using a, and also express the coordinates of point H using a.

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Chapter 4 Forms and Curves- 99
(2) The tangent line passing through the point \( (2,1) \) is not perpendicular to the $x$ axis, so its equation is \( \quad y=m(x-2)+1 \), that is $y=m x-2 m+1$. Therefore, in the equation of the line in (1), when $n=-2 m+1$,
\[m^{2}-(-2 m+1)^{2}+4=0\]
that is $3 m^{2}-4 m-3=0$. Given that lpha and eta are the two solutions of this quadratic equation, lpha and eta represent the slopes of the two tangents.
According to the relationship between roots and coefficients, \quad lpha eta=rac{-3}{3}=-1 . Therefore, the two tangents are perpendicular.
The result of (1) can be utilized.
Two lines are perpendicular $\Longleftrightarrow$ the product of their slopes is -1.

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Chapter 1 Vectors in the Plane
(3) Points C, E, and F are symmetric with respect to point B on the y-axis, origin, and x-axis, respectively, so the coordinates of points C, E, and F are
\[ \mathrm{C}(-1, \sqrt{3}), \mathrm{E}(-1,-\sqrt{3}), \mathrm{F}(1,-\sqrt{3}) \]
Also, the coordinates of point \mathrm{P} are \( (1, a) \)
Therefore $\overrightarrow{\mathrm{EP}}=\overrightarrow{\mathrm{OP}}-\overrightarrow{\mathrm{OE}}$
\[ egin{array}{l} (1-(-1), a-(-\sqrt{3})) = (2, a+\sqrt{3}) \end{array} \]
Next, since point \mathrm{H} lies on the perpendicular from point \mathrm{P} to line \mathrm{CE}, we can denote \mathrm{H}(x, a).
At this point, \( \overrightarrow{\mathrm{CH}}=\overrightarrow{\mathrm{OH}}-\overrightarrow{\mathrm{OC}}=(x-(-1), a-\sqrt{3}) \)
\[ (x+1, a-\sqrt{3}) \]
Since $\overrightarrow{\mathrm{CH}} \perp \overrightarrow{\mathrm{EP}}$, $\overrightarrow{\mathrm{CH}} \cdot \overrightarrow{\mathrm{EP}}=0$
Here, \( \overrightarrow{\mathrm{CH}} \cdot \overrightarrow{\mathrm{EP}}=2(x+1)+(a-\sqrt{3})(a+\sqrt{3}) \)
$2 x+a^{2}-1$
Therefore $2 x+a^{2}-1=0$
Consequently, x=rac{1-a^{2}}{2}
Hence, the coordinates of point \mathrm{H} are \( \left(rac{1-a^{2}}{2}, a
ight) \)

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Find the equation of the hyperbola that satisfies the following conditions: (1) the foci are at the points (3√2, 0) and (-3√2, 0), and the difference in distances from the foci is 6, (2) the foci are at the points (0, √26) and (0, -√26), and the difference in distances from the foci is 6√2

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Therefore, let point D be the point that divides side BC in the ratio 5:3, and point P lies on segment AD in the ratio 4:1.
(2) ΔPBC = (1/5)ΔABC = (2/10)ΔABC
\[
egin{array}{l}
ΔPCA = (4/5)ΔADC = (4/5) × (3/8)ΔABC = (3/10)ΔABC
\\
ΔPAB = (4/5)ΔABD = (4/5) × (5/8)ΔABC = (5/10)ΔABC
Therefore, ΔPBC:ΔPCA:ΔPAB = 2:3:5
\end{array}
\]

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Generally, prove the distance $d$ from the point \( \mathrm{P}(x_{1}, y_{1}) \) to the line $\ell: a x + b y + c = 0$ using vectors.