Plane Geometry - Properties of Basic Shapes (Points, Lines, Angles, Triangles, Quadrilaterals, Circles) | AI tutor The No.1 Homework Finishing Free App

'Find the value of k representing a circle passing through the origin (0,0).'

'Understand the explanation of distance between 1 points. Find the formulas for the distance between the origin O and point P(a), and between points A(a) and B(b).'

'(1) What kind of shape does the equation $x^{2}+y^{2}+4x-6y+4=0$ represent?\n(2) In order for the equation $x^{2}+y^{2}+2px+3py+13=0$ to represent a circle, determine the range of values for the constant $p$.'

'(1) Find the coordinates of the midpoint of the chord formed by the intersection of the line x+y=1 and the circle x^{2}+y^{2}=4, and determine the length of the chord.'

'(1) Passing through both the x-axis and y-axis, and point A(-4,2). (2) Passing through point (3,4), touching the x-axis, and having its center on the line y=x-1.'

'Plot the region that satisfies the inequalities $x^{2}+y^{2}>1$ and $|x| + |y| > 1$, and explain their relationship.'

'Example 29 | Shape of a triangle For 4 points A(4,0), B(0,2), C(3,3), D, answer the following questions.'

'Find the coordinates of the internal division points, external division points, and centroid in Example 30'

'Practice (63=>This Book p.137) (2) Let the coordinates of point P be (x, y), then from AP^2+BP^2=18 we get {(x-1)^2+(y-4)^2}+{(x+1)^2+y^2}=18, simplifying gives x^2+y^2-4y=0, which implies x^2+(y-2)^2=2^2. Therefore, the points that satisfy the condition lie on the circle (1). Conversely, any point on the circle (1) satisfies the condition. Hence, the desired locus is a circle with center at (0,2) and radius 2.'

'For the line segment connecting A (-3) and B (6), find the coordinates of the following points: (1) Point dividing internally in the ratio 2:1 (2) Point dividing externally in the ratio 2:1 (3) Point dividing externally in the ratio 1:2 (4) Midpoint'

'Taking line BC as the x-axis and point P as the origin, the coordinates of the vertices of triangle ABC can be expressed as: \nA(a, b), B(-c, 0), C(2c, 0)\nwhere b ≠ 0, c > 0. Verify the equation 2AB² + AC² = 3(AP² + 2BP²).'

'What is a diameter?'

'Find the equation of a circle that touches both the x-axis and the y-axis.'

'Find the trajectories of the following points Q and R with respect to the point P moving on the parabola y=x^2 and the two points A(3,-1), B(0,2).'

'Since the point (3,4) lies on the line 3x-2y-1=0, the desired line passes through the points (-7,-11) and (-1,6).'

'Practice (2) Parabola y=x^2 and line y=m(x+2) intersect at different points A and B. Find the locus of the midpoint of segment AB as the value of m varies.'

'Find the standard equation of a circle.'

'The equation of the line passing through the points (-1,1) and (3,-1) is $y-1=\\frac{-1-1}{3-(-1)}(x+1)$, which simplifies to $x+2y=1$.'

'When the chapter 3 (28 t) takes all real values, for the three points A(t, t^{2}), B(t, t-2), C(t+√3, t^{2}-t-1), answer the following questions:\n(1) Prove that for each real number t, A and B are distinct points.\n(2) Find all t values that make triangle ABC a right triangle.\n(3) Determine the range of t values that make triangle ABC an acute triangle.'

'The equation of the perpendicular bisector of line segment BC is y-0=-2(x-5), which simplifies to y=-2x+10. By solving equations (4) and (5) simultaneously, we get x=4, y=2. Therefore, the center of the circumscribed circle is at point (4,2) and the radius is sqrt{(8-4)^{2}+(5-2)^{2}}=5. Thus, the equation we are looking for is (x-4)^{2}+(y-2)^{2}=25.'

'Circle with center (-2, 3) and radius 3'

'When the intersection points of 22 circles, a circle passing through the intersection points of a circle and a line, and the equations of lines in terms of x, y are written as f(x, y), the curve represented by the equation f(x, y) = 0 (including cases where it represents a line) is called the curve f(x, y) = 0 and the equation is called the equation of the curve.'

'Find the equation of the tangent line at point P(4,6) on the circle.'

'Find the values of the constant k for which the lines do not form a triangle.'

'The results from (1) to (3) indicate that the coordinates of the centroid G of △PQR change from $\\left(\\frac{3-8+2}{3}, \\frac{3+1+1}{3}\\right)$ to $\\left(-1, \\frac{5}{3}\\right)$.'

'(2) Find the locus of the point P for which the sum of the squares of the distances from the points A(1,4) and B(-1,0), i.e., AP^2 + BP^2 is 18. [(2) Hokkai Gakuen University]'

'Given that the length of the perpendicular dropped from point (2,1) to the line kx + y + 1 = 0 is √3, find the value of the constant k.'

'Find the equations of the lines parallel and perpendicular to the line 4x+3y-6=0 passing through the intersection point of the lines 2x-y-1=0 and x+5y-17=0.'

'Intersect at points (-1, 3) and (3, -1)'

'Find the distance between the following two points.'

'(2) (1) From (1), in triangle △AOB where ∠AOB = 90°, the circle passing through points A, B, O has AB as its diameter.'

'Practice Example 17 Area of a Sector'

'Find the arc length and area of a sector with a radius of 4 and a central angle of 150°.'

'Problem (1) In the coordinate plane, when the points A(a, 2), B(5, 1), C(-4, 2a) are collinear, find the value of the constant a.'

'(1) It touches both the x-axis and the y-axis, passing through point A(-4,2). (2) Passing through point (3,4), touching the x-axis, with the center on the line y=x-1.'

'Problem to find the coordinates of point P. Find the coordinates of point P (x, y) lying on the line connecting two points A (6, -3) and B (1, 7).'

'Outside of the circle (x-2)^{2}+(y-1)^{2}=5. Including the boundary.'

'The coordinates of the centroid G of triangle ABC with vertices A(x1, y1), B(x2, y2), and C(x3, y3) are (\\frac{x1+x2+x3}{3}, \\frac{y1+y2+y3}{3})'

'Maths II river 36 books p.119\n(1) The radius r is the distance between the center (-5,4) and the origin, so r^2=(-5)^2+4^2=41\nTherefore, the equation of the circle we seek is (x+5)^2+(y-4)^2=41\n(2) The center is the midpoint of the diameter, so its coordinates are (-3+3)/2, (6+(-2))/2 which is (0,2)\nThe radius r is the distance between the center (0,2) and the point A(-3,6), so r^2=(-3-0)^2+(6-2)^2=25\nTherefore, the equation of the circle we seek is x^2+(y-2)^2=25\nAnother solution (2) On the circumference, let P(x, y) be a point different from A, B, then AP ⊥ BP so, when x ≠ -3, x ≠ 3, (y-6) / (x-(-3)) * (y-(-2)) / (x-3) = -1\nTherefore, (x+3)(x-3)+(y-6)(y+2)=0 which is x^2+(y-2)^2=25\nThis equation holds when x=-3, x=3, i.e., the points (-3,6), (-3,-2), (3,6), (3,-2) satisfy it, so this is the equation of the circle we seek.'

'For a standard angle, determine whether the following proposition is true and explain why.\n"There are no angles greater than 360 degrees"'

'Taking OP as the radius.'

'(5) A line parallel to the y-axis is perpendicular to the x-axis. Since the x-coordinate of the point it passes through is 5, we have x=5'

'If a parabola and a circle are tangent at one point, as shown in the figure, if the parabola and circle are tangent at the point (0,3) or the point (0,-3), then a = ±3, so the desired value of a is a = -37/4, ±3.'

'Important Example 58: Intersection Points of Parabola and Circle\nLet r be a positive constant. Consider the parabola y=x^{2} and circle x^{2}+(y-2)^{2}=r^{2}, and answer the following questions:\n(1) When r=2, find all coordinates of intersection points between the parabola and the circle.\n(2) Investigate how the number of intersection points between the parabola and the circle changes as r varies over all positive real values.'

'Find the values of a when the lines (a-2)x+ay+2=0 and x+(a-2)y+1=0 are parallel, coincident, or perpendicular.'

'Understand the formula for the distance between 2 points in a plane. Find the formula for the distance between points O(0; 0), A(x_{1}, y_{1}), B(x_{2}, y_{2}).'

'For the two circles \ x^{2}+y^{2}-2 x-4 y+1=0, x^{2}+y^{2}=5 \:\n(1) Find the equation of the line passing through the two intersection points of the two circles.\n(2) Find the center and radius of the circle passing through the two intersection points of the two circles and the point (1,3).'

'Which of the following lines are parallel to each other, and which are perpendicular?\n(1) y=2x+3\n(2) y=√2x-1\n(3) y=-2x+1\n(4) 2x-√2y+1=0\n(5) x+2y-5=0'

'(1) Find the equation of a circle with center (-5,4) passing through the origin.\n(2) Find the equation of a circle with diameter AB where A(-3,6) and B(3,-2).'

'Given circles, denoted as C_1 and C_2, respectively. (1) Let the coordinates of the point of tangency on circle C_1 be (x_1, y_1) such that x_1^2 + y_1^2 = 9'

'Important Example 49 Distance between a point on a parabola and a line\nGiven two points A(0,1) and B(2,5) and the parabola y=x^{2}+4x+7. Let there be a moving point P on the parabola.\n\nFind the minimum value of the area S of triangle PAB.'

'Plot the range of existence of point (a, b) on the ab plane when the line segment connecting points A(1,-2) and B(-2,1) intersects the parabola y=x^{2}+ax+b at only one point other than A and B.'

'Calculate the distance between two points on a plane.'

'The equation of the line AB is x/a + y/b = 1. Let point P(a, b) and the distance between point P and the line AB be d. Find the maximum value of d.'

'Number of lattice points'

'Let the equation of the required circle be (x-1)^(2)+(y+√3)^(2)=r^(2) (r>0). The condition for the circle (2) to be tangent to circle C is 0<r<5 and √((1-0)^(2)+(-√3-0)^(2))=5-r, therefore r=5-√4=3. Hence, the required equation is (x-1)^(2)+(y+√3)^(2)=9'

'Find the coordinates of point Q:\n\nLet the coordinates of point Q be (x, y).\n(1) Let OP=r, and let the angle between OP and the positive direction of the x-axis be α, then r cosα=-2, r sinα=3.\nTherefore, x=r cos(α+5/6π)=r cosα cos5/6π-r sinα sin5/6π.'

'Coordinates of points\nLet the points A(x₁, y₁), B(x₂, y₂), C(x₃, y₃) be given.\nFind the distance between two points.\nAB=√((x₂-x₁)² + (y₂-y₁)²)\nIn particular, the distance between the origin O and A is OA=√(x₁² + y₁²)'

'Example 55 Tangent Conditions and Equations of Circles and Lines'

'Taking point B as the origin and edge BC as the x-axis, the coordinates of each vertex can be represented as A(0, a), B(0, 0), C(b, 0), D(b, a). Prove that PA² + PC² = PB² + PD².'

'On a plane, there are n circles such that any two circles intersect each other and no three or more circles intersect at the same point. How many parts does the plane get divided into by these circles?'

'Passing range of points and curves'

'Similar question Taking a point P(1/2, 1/4) on the coordinate plane. When two points Q(α, α^2) and R(β, β^2) on the parabola y=x^2 move such that the three points P, Q, R form an isosceles triangle with QR as the base, find the locus of the centroid G(X, Y) of triangle PQR. [University of Tokyo]'

'The parabola and the circle have 4 shared points when the vertex of the parabola is on the line segment connecting point (0, -37/4) and point (0, -3) (excluding the endpoints) as shown in the figure. Therefore, -37/4 < a < -3.'

'Two circles tangent to the coordinate axes and a line'

'Taking the line BC as the x-axis and the perpendicular bisector of side BC as the y-axis, the midpoint L of side BC becomes the origin O, and the coordinates of each vertex can be represented as A(a, b), B(-c, 0), C(c, 0). In this case, L(0,0), M((a+c)/2, b/2), N((a-c)/2, b/2), therefore, the coordinates of the intersection points dividing the three medians AL, BM, CN in a 2:1 ratio are ((a/3), (b/3)), ((-c+(a+c))/(2+1), (0+b)/(2+1)), ((c+(a-c))/(2+1), (0+b)/(2+1)), all of which are ((a/3), (b/3)), so the three medians AL, BM, CN intersect at this point.'

'Practice 1: Find the radius and total area of the incircles of equilateral triangles'

'Illustrate the radius of the following angles. Also, identify the quadrant in which they lie.'

'(1)\n{% raw %}\\(\\mathrm{AB}^{2}=(0-4)^{2}+(2-0)^{2}=20\\)\\(\\mathrm{BC}^{2}=(3-0)^{2}+(3-2)^{2}=10\\)\\(\\mathrm{CA}^{2}=(4-3)^{2}+(0-3)^{2}=10\\)\\{% endraw %}\nTherefore, BC=CA, BC^2 + CA^2 = AB^2, so ΔABC is a right-angled isosceles triangle with ∠C=90∘.'

'Find the equation of a line'

'Practice Let real number t satisfy 0<t<1, consider the 4 points O(0,0), A(0,1), B(1,0), C(t,0) on the coordinate plane. Also, define point D on segment AB such that ∠ACO=∠BCD. Find the maximum area of triangle ACD. [University of Tokyo]'

'When the point (x, y) moves inside a circle with radius 1 centered at the origin, depict the range of motion of the point (x+y, x y).'

'For the circle $C: x^{2}+y^{2}=25$, answer the following questions:\n1. Find the equation of a circle with center at $(12,5)$ that is tangent to circle $C$ externally.\n2. Find the equation of a circle with center at $(1,-\\sqrt{3})$ that is tangent to circle $C$ internally.'

'Find the arc length and area of a sector with radius 4 and central angle 150 degrees.'

'Let a and b be positive real numbers. The parabolas C1: y = x^2 - a and C2: y = -b(x - 2)^2 are both tangent to the line ℓ at the point P(x0, y0). Define S1 as the area enclosed by the line x = 0, the parabola C1, and the tangent line ℓ, and S2 as the area enclosed by the line x = 2, the parabola C2, and the tangent line ℓ. Answer the following questions:\n(1) Express a, x0, y0.\n(2) Express the ratio of areas S1 : S2 in terms of b.'

'Distance between a point and a line'

'Represent the set of points (x, y) that satisfy y=x+1 in a figure with a straight line as the boundary. Also, represent the regions of points that satisfy y>x+1 and y<x+1 in a figure.'

'Find the coordinates of a point P on the x-axis equidistant from points A(-1,2) and B(3,4).'

'Find the equation of a circle passing through the point (2,1) and tangent to the x-axis and y-axis.'

'Basic Example 70: Given A(-2,1), B(6,-3), C(1,7), find the coordinates of the following points.'

'Consider the following problem.'

'Investigate how the number of intersection points between the circle (x-1)^2+(y-1)^2=r^2 and the line y=2x-3 changes depending on the radius r.'

'Standard Example 103'

'Given the three vertices A(5,-2), B(1,5), C(-1,2), find the lengths of the three sides of triangle ABC and determine what type of triangle it is.'

'Plot the region represented by the following inequalities.'

'Find the equation of the tangent line drawn from point A(3,1) to the circle x^2+y^2=2 and the coordinates of the point of tangency.'

'Let A and B be the intersection points of the parabola y=9-x^{2} and the x-axis. When a trapezoid is inscribed in the area enclosed by this parabola and the x-axis, with segment AB as the base, determine the maximum area of this trapezoid.'

'Find the number of lattice points within the area enclosed by y = -x^2 + 8x and y = x (including the boundary).'

"A circle with center C(a, b) and a constant distance r(>0) from C is a collection of points with C as the center and radius r. The circle with center C is simply called circle C, and the equation satisfied by any point (x, y) on the circle is called its equation. Let's try to find the equation of this circle. The condition for a point P(x, y) to be on circle C is CP = r, expressed in coordinates as √((x-a)^2 + (y-b)^2) = r, squaring both sides gives (x-a)^2 + (y-b)^2 = r^2. Since both sides of (2) are positive, (1)⇔(2)⇔(3), hence (3) is the equation of the desired circle. The form of equation (3) with knowledge of the center (a, b) and radius r is called the basic form of the circle equation. The equation of a circle with radius r and center (a, b) is (x-a)^2 + (y-b)^2 = r^2. The equation of a circle with radius r and center at the origin is x^2 + y^2 = r^2. Note that setting a=b=0 in 1 results in 2. When r=1, it is called a unit circle. Additionally, 1 can be considered as a translation of 2 parallel to a along the x-axis and b along the y-axis."

'(1) Let the circle (x-1)^2 + (y+2)^2 = 9 be C. When the circle (x+1)^2 + (y-1)^2 = 4 is C₁, determine the position relationship between C and C₁.'

'Point D is in the fourth quadrant, Circle D is tangent to the x-axis and y-axis, so the coordinates of point D can be assumed as (d, -d) and the radius is d. Since point D is below the line l, we have 3d - 4d - 12 < 0. The distance between point D and line l is |3d - 4d - 12| /√(3^2+4^2) = (d + 12) / 5. Since Circle D is tangent to line l, the distance between point D and line l is (d + 12) / 5 = d. Therefore, d = 3.'

'Find the equation of the tangent line at point A on the circle with A(0,3) and B(8,9) as the diameter.'

'There are points A(-1,3) and B(5,11).'

'Find the value of a when the two circles touch each other.'

'Find the equation of a line that is tangent to the circle x^2 + y^2 = 9 and parallel to the line 4x + 3y - 5 = 0.'

'For points A(0,1) and B(4,-1): (1) Find the equation of a circle C1 with center on the line y=x-1 passing through points A and B. (2) Find the equation of a circle C2 which is symmetric to the circle C1 found in (1) with respect to the line AB. (3) Let P and Q be points on circles C1 and C2, respectively. Find the maximum length of the line segment PQ. [Gunma University]'

'There is a parallelogram ABCD with vertices A(-2,3), B(5,4), and C(3,-1). Find the coordinates of vertex D and the intersection point P of the diagonals.'

'Find the equation of the following circle:\n(1) A circle with center at (1,1) that is tangent to the line 2x-y-11=0'

'Given points A(6,0) and B(3,3), when point P moves on the circle x^2+y^2=9, find the locus of the centroid G of triangle ABP.'

"Let C and C' be the two intersection points A, B, and the midpoint of line segment AB be M. Therefore, the length of segment OM is equal to the distance between the origin O and the line ℓ."

'Using the coordinates (p, q) of point B, determine the condition for the line AB to be perpendicular to the line ℓ when the slope of ℓ is 2.'

'When exactly 2 tangent lines can be drawn from point P(1, ) to curve C, answer the following questions. (i) Find the equations of the 2 tangent lines. (ii) Let Q and R be the points of tangency between the lines found in (i) and curve C. Assume that the x-coordinate of Q is less than the x-coordinate of R. Find the area S of the figure enclosed by line segment PQ, line segment PR, and curve C.'

'Investigate the positional relationship between the following circles and lines, and find the coordinates of the intersection points if they exist.'

'What kind of shapes do the following equations represent?'

'Assume there are four circles on the coordinate plane that touch the x-axis, y-axis, and the line 3x + 4y - 12 = 0. Arrange the radii of these circles in ascending order and explain the relationship between the center of each circle and the line.'

'Find the slope of the line that makes an angle of \\frac{\\pi}{4} with the line x - \\sqrt{3} y = 0.'

'For the point A(-2, -3), find the coordinates of point Q which is symmetrical to point P(3, 7).'

'Find the equation of the tangent line at point P on the following circle.'

'Let A be the intersection of the two lines \ 3 x+2 y-4=0 \ (1) and \ x+y+2=0 \ (2). Determine the equation of the line passing through point A and B(3,-2) for (1). Determine the equation of the line passing through point A and parallel to the line \ x-2 y+3=0 \ for (2).'

'Given A(-2,-3), B(3,7), C(5,2), find the coordinates of the following points.'

"Let's find the equation of the tangent line at point (a, b) on the circle x^2 + y^2 = r^2."

'Let line 3x+2y-4=0 be (1) and x+y+2=0 be (2), with A as the intersection point of the two lines. Find the equation of the line passing through A and point B(3,-2).'

'Find the equation of the line that intersects the x-axis at (2, 0) and the y-axis at (0, -3).'

'What is the shape of the triangle ABC formed by the following 3 points?'

'Find the equations of the following circles:\n1. Circle with center at (2, -3) and radius 1\n2. Circle with center at (3, 4) passing through the origin\n3. Circle with diameter defined by the points (3, 1) and (-5, 7)\n4. Circle with center at (5, 2) tangent to the y-axis'

'Find the center and radius of the circle that passes through the two intersection points of the two circles \\( x^{2}+y^{2}=2,(x-1)^{2}+(y+1)^{2}=1 \\) and is tangent to the line \ y=x \.'

'Point A is in the first quadrant, Circle A is tangent to the x-axis and y-axis, so the coordinates of point A can be assumed as (a, a), with the radius being a. Since point A is below line l, we have 3a + 4a - 12 < 0. The distance between point A and line l is |3a + 4a - 12| / √(3^2 + 4^2) = (-7a + 12) / 5. Since circle A is tangent to line l, the distance between point A and line l is a, hence (-7a + 12) / 5 = a, leading to a = 1.'

'Master the formulas for internal and external division points coordinates and conquer example 74!'

'Circle and line'

'For which values of the constant k does the circle C: x^2+y^2+(k-2)x-ky+2k-16=0 pass through the points A(x, y) and B(x, y)? Here, . The line segment AB will be a diameter of the circle C only when k=.'

'Explain what the third quadrant is.'

'Show the region boundaries represented by the inequalities.'

'Problem to find the position relationship between a line and a circle, along with the coordinates of their intersection points.'

'Find the equations of the following circles.'

"The equation of the line passing through the two intersection points C and C' is \\square x+\\square y=15. Also, the area S of the triangle with the two intersection points and the origin O as vertices is S=\\square."

'What kind of shapes do these equations represent?'

'Are the following two lines parallel or perpendicular?'

'Master the formula for distance between a point and a line, conquer example 83!'

'Illustrate the radius of angle θ and specify in which quadrant the angle lies'

'When the triangle ABC is a right triangle with vertices A(1,1), B(2,4), and C(a,0), find the value of the constant a.'

'Find the coordinates of the point P on the circumference of the circle with equation x^2-2x+y^2-4y+4=0 that is closest to the point A(-1,1). Also, find the distance between the points A and P.'

'Point on a plane'

'(2) Right isosceles triangle where \ \\angle \\mathrm{A}=90^{\\circ} \'

'Regarding points A(0,1) and B(4,-1)'

'Chapter 4: Figures and Equations'

'Problem of finding the distance between two points on a plane.'

'Find the coordinates of the following points.'

'Given the circle TR: x^{2}+y^{2}=1, denoted as C_{0}, and let C_{1} be the circle obtained by translating C_{0} 2a units in the positive direction of the x-axis, where a is 0<a<1. Also, let A and B be the two intersection points of C_{0} and C_{1} in the first quadrant, and let P(s, t) be a point on C_{0} different from points A and B. Find the locus of the centroid G of triangle PAB as P moves on the part of C_{0} excluding the two points A and B.'

'73 (1) $\\mathrm{AB}=\\mathrm{CA}$ is an isosceles triangle'

"The position of point P on a plane is represented by a pair of real numbers, for example, (a, b). This pair (a, b) is called the coordinates of point P, where a is the x-coordinate and b is the y-coordinate. The point P with coordinates (a, b) is denoted as P(a, b). In this section, let's learn about points on a plane. The plane with coordinates is divided into 4 parts by coordinate axes. These parts are called quadrants, and they are named as the first quadrant, the second quadrant, the third quadrant, and the fourth quadrant in a counterclockwise manner. Note that the coordinate axes are not included in any quadrant. In the diagram, (+, +) indicates the signs of x and y coordinates in each quadrant."

'In this case, the distance between the center (0,0) of circle (1) and the line (2) is equal to the radius of the circle √k, so'

'Find the number of tangents that can be drawn from point P(1, ) to the curve C: y=x^3-x.'

''

'Given the center and radius of a circle, find the equation of the circle.'

'Find the equation of a circle with center (a, b) and radius r.'

'Find the circle passing through the intersection of 2 circles'

'Prove that for an acute triangle ABC, the following equation holds: tan A + tan B + tan C = tan A tan B tan C.'

'Find the locus of the midpoint P of the line segment connecting point A(2,0) and point Q as point Q moves along the circle x^2 + y^2 = 1.'

'Examine the position relationships between the following circles and lines, and if there are common points, find their coordinates.'

'Since the center of circle C3 is the origin O, the distance between circle C and circle C3 is PO=√(1^2+(-2)^2)=√5\nLet r3 be the radius of circle C3, as circle C3 is inscribed in circle C, we have r3 < 3 and √5 = 3 - r3\nTherefore r3 = 3 - √5\nHence, the equation of circle C3 is x^2 + y^2 = (3 - √5)^2'

'Plot the regions represented by the following inequalities.'

'There is a parallelogram ABCD with vertices A(-2,3), B(5,4), and C(3,-1). Find the coordinates of vertex D and the intersection point P of the diagonals.'

'Find the coordinates of the midpoint and the length of the segment cut by the circle with center (2, 1) and radius 2 from the line y=-2x+3.'

'Find the equation of the tangent line drawn from point A(7,1) to the circle x^2+y^2=25.'

'Find the equation of a circle passing through the points (0,2) and (-1,1) with its center on the line y=2x-8.'

'When a parabola (1) and a circle (2) have 4 common points, find the range of r.'

'Find the center and radius of the circle passing through the two intersection points of the two circles x^2+y^2=2 and (x-1)^2+(y+1)^2=1, which is tangent to the line y=x.'

'In Figure 6, what is the length that can be read with the vernier calipers in increments of how many millimeters?'

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'What are the values on the vertical axis for points (2) and (3)?'

'2021 Shibuya Academy Makuhari Middle School (1st time) (4)\nAs shown in Figure 5-1, there is a rectangular prism $ABCD-EFGH$ with a rhombus base and all rectangular side faces. Points $K, L, M, N$ are located on the edges $AB, BC, CD, DA$ respectively, with $AK: KB = AN: ND = 1: 1$, $BL: LC = DM: MC = 2: 1$.\nAdditionally, point O is located on the diagonal $EG$ of the rhombus $EFGH$, with $EO: OG = 1: 5$.\nConnecting each vertex of quadrilateral $KLMN$ to point O creates pyramid O-KLMN. Answer the following questions. The volume of the pyramid can be calculated as (base area) x (height ÷ 3).'

'Explain the difference between a bright red star and a dark red star.'

'There is a right triangle ABC as shown in the figure 2, and squares with sides of AD, BD, and CD respectively. In this case, what is the area of the square with CD as one side in square centimeters?'

'What is the size of cell A in figure 12 (length between PQ) in micrometers? Use the value obtained in (5) and answer in integers.'

'peninsula'

'(3) As shown in the graph on the right, cliff B is 48m above sea level and at a distance of 70m north from point A, cliff C is 53m above sea level and at a distance of 70m south from point A, simply note down their respective positions.'

'(2) The line drawn by point O becomes like a bold line. Firstly, the central angle of a semi-circle with a radius of 6 cm is between (2) and (3). Adding the part between 8 and 9 (which is equal to the length of a 60-degree arc) gives a total of 180×3+90+60×2 = 750 degrees. Moreover, the arc between 3 and 4 has a radius of 12+6=18 cm and a central angle of 30 degrees. Therefore, the length of the line drawn by point O is calculated as 6×2×3.14×750/360+18×2×3.14×30/360=(25+3)×3.14=87.92 cm.'

'In the figure 5-1, there is a right-angled triangle with angle A, where AB=3 cm and AC=6 cm, and a right-angled isosceles triangle with angle D, where DE and DF are both 6 cm. Answer the following questions about the geometric shape formed by combining these right-angled triangles. Take the value of Pi as 3.14. Also, note that the volume of a cone can be calculated by base area * height / 3.'

'The height above sea level of volcanic ash layer X is 53 meters at cliff A and 44 meters at cliff B. When marked with circles on a graph, it looks as shown in the right graph.'

'(4) A set of points where the difference in distance from A to the sound source and B to the sound source is constant at 350m forms a line, indicating the presence of the sound source. This is represented by (I). A curve where the difference in distance from two points to the sound source is constant is called a hyperbola.'

'Does not fit in A3 paper'

"Comets are celestial bodies in the solar system that revolve around the sun like planets. Comets exhibit a distinctive feature of suddenly brightening as they approach the sun from far away in the solar system and darkening abruptly and disappearing as they move away. Additionally, as shown in Figure 6, comets present a different appearance with a long tail waving unlike other celestial bodies. The tail of a comet extends in the opposite direction of the sun. (5) Suppose a new comet is discovered, and it is visible immediately after the sunset of that day. Describe the appearance of the comet's tail as a straight line."

'(3) The sound source A is at the position reached within 1 second, and B is at the position reached within 2 seconds. Therefore, drawing a circle with a radius of 350 × 1 = 350 meters centered at A, and a circle with a radius of 350 × 2 = 700 meters centered at B, the two intersection points of these two circles will indicate the position of the sound source.'

'Choose one correct statement that can be inferred from the graph in Figure 4.'

'Find the length of each side of the following triangle.'

'Equation of a plane perpendicular to two coordinate axes'

'Right triangle with angle C equal to 90 degrees'

'Let the polar coordinates of point A be (r₁, θ₁) and those of point B be (r₂, θ₂). Find the area of triangle OAB, denoted by S.'

'In parallelogram ABCD, let M be the midpoint of side AB, E be the point that divides side BC into 1:2, and F be the point that divides side CD into 3:1. If →AB=b and →AD=d'

'A regular hexagon ABCDEF with side length 1 is given. When point P moves along side AB and point Q moves along side CD independently, find the area where point R, which divides segment PQ in the ratio 2:1, can pass through.'

'Although it is possible to directly substitute z=x+yi into the equation (3)(2) and calculate it, the computation becomes very complicated (see the first consideration after the answer). Therefore, we first consider the equation of an ellipse with congruence to shape C and having the focus on the x-axis, then rotate it to find the equation of C. (1) K: \\frac{x^{2}}{2^{2}}+\\frac{y^{2}}{1^{2}}=1 The coordinates of the foci are, \\sqrt{2^{2}-1^{2}}=\\sqrt{3}, so it is (\\sqrt{3}, 0),(-\\sqrt{3}, 0). The length of the major axis is 2\\cdot2 and the length of the minor axis is 2\\cdot1, therefore, the area to find is \\pi\\cdot2\\cdot1=2\\pi.'

'The tangent at any point on the curve C in the first quadrant always intersects the positive parts of the x-axis and y-axis, and the intersection points are denoted as Q and R, respectively. The point P divides the segment QR internally in the ratio 2:1.'

'For polar coordinates, find the equations of the following circle and line: (1) A circle with center at point A(3, π/3) and radius 2. (2) A line passing through point A(2, π/4) and perpendicular to OA (O is the pole).'

'(2) 128\nIn the isosceles trapezoid \ \\mathrm{ABCD} \ where \ \\mathrm{AB}=2 \\mathrm{~cm}, \\mathrm{BC}=4 \\mathrm{~cm}, \\angle \\mathrm{B}=60^{\\circ} \, when \ \\angle \\mathrm{B} \ increases by \ 1^{\\circ} \, by how much does the area \ S \ of the trapezoid \ \\mathrm{ABCD} \ increase? Assume \ \\pi=3.14 \.'

'Find the distance between two points.'

'Find the position vector of the orthocenter of a triangle.'

'In the coordinate space, let A(1,0,2) and B(0,1,1). When point P moves along the x-axis, find the minimum value of AP+PB.'

'On the coordinate plane, the circle C passes through the point (0,0), its center lies on the line x+y=0, and it is tangent to the hyperbola xy=1. Find the equation of circle C. Here, the circles and hyperbolas are said to touch at a point if the tangents of the circle and the hyperbola coincide at that point.'

'For the ellipse $x^{2}+4 y^{2}=4$, find the locus of the point $P(a, b)$ outside the ellipse from which two tangents drawn to the ellipse intersect at right angle.\n[Type University of Tokyo]\nBasic 155'

'Curve represented by parametric variables'

'Find the geometric figure represented by all points P(z) that satisfy the equation $|z-\\alpha|=r(r>0)$.'

'On the curve \\sqrt[3]{x}+\\sqrt[3]{y}=1, let \\mathrm{P} be the point in the first quadrant where the tangent intersects the x-axis and y-axis at points \\mathrm{A}, \\mathrm{B} respectively. If the origin is \\mathrm{O}, find the minimum value of \\mathrm{OA}+\\mathrm{OB}.'

'Find the coordinates and length of the chord formed by the intersection of the following line and curve.'

'Basic Concepts 1 Polar and Straight-line Equations (1) Polar equation of a circle with center at the pole O and radius a r=a r=2a cos θ r^2-2r r₀ cos(θ-θ₀)+r₀^2=a^2 θ=α r cos (θ-α)=a (a>0) (2) Circle with center at (a, 0) and radius a r=2a cos θ (3) Circle with center at (r₀, θ₀) and radius a r^2-2r r₀ cos(θ-θ₀)+r₀^2=a^2 (4) Line passing through the pole O and forming an angle α with the initial line θ=α (5) Line passing through point A(a, α) and perpendicular to OA'

'Let the equation $2 x^{2}-8 x+y^{2}-6 y+11=0$ represent a conic section $C_{1}$. Also, let $a, b, c$ $(c>0)$ be constants, and the equation $(x-a)^{2}-\\frac{(y-b)^{2}}{c^{2}}=1$ represent a hyperbola $C_{2}$. Determine the values of $a, b, c$ when the two foci of $C_{1}$ and the two foci of $C_{2}$ form the 4 vertices of a square.'

'Using the 4 points given in question (1) A(2,4), B(-3,2), C(-1,-7), D(4,-5), form a quadrilateral\n(2) Using the 3 points A(0,2), B(-1,-1), C(3,0) as vertices, connect another point D to form a parallelogram. Find the coordinates of the fourth vertex D.'

'(4) For the plane PQR and edge OD, the situations are as follows. When q = 1/4, plane PQR is . When q = 1/5, plane PQR is 又. When q = 1/6, plane PQR is ネ. Choose the one that fits two 〜 ネ, one from 0 to 5, you can choose the same option repeatedly.'

'Find the locus of the center P of a circle that is tangent to both the circle $(x-4)^{2}+y^{2}=1$ and the line $x=-3$.'

'For the ellipse \\( \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a>b>0) \\), the coordinates of the foci are \\(\\left(\\sqrt{a^{2}-b^{2}}, 0\\right),\\left(-\\sqrt{a^{2}-b^{2}}, 0\\right)\\). The foci are on the x-axis, with the length of the major axis being 2a and the length of the minor axis being 2b.'

'Prove that the difference in distance from any point on the hyperbola to its two foci is constant.'

'State the condition for three distinct points A(α), B(β), C(γ) to be collinear.'

'Let the two endpoints A and B of a line segment of length 2 move along the x-axis and y-axis, respectively. When $\\mathrm{BP}=1$, find the trajectory of point P.'

'Passing through the point A(3, -4), find the line parallel to the line l: 2x-3y+6=0 and name it g. Determine the equation of the line g.'

'In the regular hexagon ABCDEF, with center O, point P divides side CD internally in the ratio 2:1, and point Q is the midpoint of side EF. If vector AB is a and vector AF is b, express vectors BC, EF, CE, AC, BD, QP in terms of vectors a and b.'

''

'When a point P(z) moves along the perimeter of a circle centered at -i with a radius of 1 (excluding the origin), the point Q(w) represented by (3) 114 w=1/z, what kind of shape does it trace?'

"Topic: Investigation of quadratics represented by complex number equations and rotation movement Mathematics In Chapter 3 of mathematics C, we learned about geometric shapes in the complex plane, and in Chapter 4 we learned about the properties of quadratics. Here, we will investigate cases where the shape represented by the complex number z's equation is a quadratic. First, let's confirm the basic concepts of quadratics with the following problem. CHECK 3-A The equation of the trajectory of point P with a total distance of 6 from points F(√5, 0) and F'(-√5, 0) is to be found."

'Find the vector equation of a circle.'

'Plot the region E on the ab-plane consisting of all points (a, b) where the hyperbola $\\frac{x^{2}}{4}-\\frac{y^{2}}{9}=1$ and the line $y=ax+b$ have common points.'

'Assume that △ABC is an equilateral triangle with vertices A(-1), B(1), and C(√3 i). Prove that when △PQR with vertices P(α), Q(β), R(γ) is also an equilateral triangle, the equation α² + β² + γ² - αβ - βγ - γα = 0 holds true.'

'Prove the equation of the tangent line at the point (x0, y0) on the circle using vectors.'

'In triangle △OAB with vertices O(0), A(1), B(ι), where ∠O is the right angle vertex, prove that when triangle △PQR is formed by points P(α), Q(β), R(γ) with ∠P as the right angle vertex, the equation 2α² + β² + γ² - 2αβ - 2αγ = 0 holds.'

'(2) The sum of the distances from point z to the 2 points (√3+3i)/2 and -(√3+3i)/2 is constant at 4, therefore, the figure C is an ellipse with the 2 foci at (√3+3i)/2 and -(√3+3i)/2. Let c be the distance from the origin, which is the center of this ellipse, to the foci. The coordinates of the foci on the xy-plane are (c, 0) and (-c, 0). This ellipse is congruent to an ellipse where the sum of the distances from points on the ellipse to the 2 foci is also 4.'

'Consider a circle C with radius a in the first quadrant of the xy-plane, which is tangent to both the line l: y=mx(m>0) and the x-axis. Also, consider circles tangent to the line l, the x-axis, and the circle C at one point each with a radius of b, where b>a. (1) Express t in terms of m. (2) Express b/a in terms of t. (3) Find the limit lim_{m \to +0} 1/m(b/a-1).'

'(1) Let the lengths of the three sides of triangle ABC be AB=8, BC=7, CA=9. Let vector AB=b and vector AC=c, and let P be the incenter of triangle ABC. Express vector AP in terms of b and c.'

'Midpoint theorem: In triangle ABC, let the midpoints of sides AB and AC be M and N respectively. Then MN // BC and MN = 1/2 BC'

'There is a quadrilateral ABCD inscribed in a circle. When AB=4, BC=5, CD=7, DA=10, find the area S of quadrilateral ABCD.'

'Circumcenter of a triangle'

'Determine the range of values for x so that a triangle with side lengths 3, 5, and x becomes an acute triangle.'

'A regular tetrahedron OABC with edge length of 6 is given. Let L be the midpoint of edge OA, M be the point that divides edge OB into 2:1, and N be the point that divides edge OC into 1:2. Find the area of triangle LMN.'

'In triangle ABC, if B=30°, b=√2, and c=2, find the values of A, C, and a.'

'Please provide the terms related to the circumcircle of a triangle and their definitions.'

'The center of the circle passing through points A, P, and Q is the point of intersection of the perpendicular bisectors of two chords.'

'(2) acute angle'

'Using the power of a point theorem, show the properties of the two tangents drawn from point P to a circle.'

'For a regular octagon, find the following numbers.\n(1) The number of quadrilaterals that can be formed by connecting 4 vertices\n(2) The number of triangles formed by connecting 3 vertices that share an edge with the regular octagon'

'On the line x=1, the point T is located where the y-coordinate is √3. The point P is the intersection of the line OT and a semicircle with radius 1. The angle we are looking for is ∠AOP.'

'On a rectangular floor with dimensions of 240 cm by 396 cm, we want to cover it with square tiles of side length a cm without any gaps. Find the maximum value of a in this case. Also, determine the number of tiles that can be laid out.'

'Explain and prove the properties of the circumcenter, incenter, and centroid of a triangle.'

'Determine whether the four points A, B, C, D on the right diagram lie on the same circle.'

'The radius of circumcircle is \ \\frac{85}{8} \, and the radius of incircle is 2'

'In a quadrilateral ABCD inscribed in a circle, with AB = 8, BC = 10, and CD = DA = 3. Find the area S of the quadrilateral ABCD.'

'79. In triangle ABC, AB=2, BC=4, CA=2√3. Let AD be the altitude from vertex A to side BC, and let E and F be the points of intersection of the circle with diameter AD and sides AB and CA, respectively. E and F are different from A. [Tokyo Jikeikai Medical University]\n(1) Prove that points E, B, C, and F lie on the same circle.\n(2) Find the area of triangle EBF.'

'As shown in the diagram, label all vertices of an equilateral triangle with side length 2 and each midpoint of the edges from 1 to 6. Match the outcome of the first dice roll with this number. Connect the points corresponding to the numbers rolled on the dice three times to create a shape. Find the expected value of the area of the resulting shape.'

'Theorem 25 Power of a Point Theorem II'

'In \ \\triangle ABC \, with \ \\angle A=90^{\\circ}, \\angle B=60^{\\circ}, \\angle C=30^{\\circ} \ and noting that \ AD \ is the diameter of a circle, draw the auxiliary lines \ AD, ED, EF, DF \.'

'Let p and q be the lengths of the diagonals AC and BD of quadrilateral ABCD, and let θ be one of the angles formed by the diagonals. Express the area S of quadrilateral ABCD in terms of p, q, and θ.'

'Please explain about points inside and outside a circle and the sizes of angles.'

'In triangle ABC, O is the circumcenter. Find the angles α and β in the figure on the right.'

'Conditions for a parallelogram: A quadrilateral is a parallelogram if any of the following conditions are met. [1] Two pairs of opposite sides are parallel. [2] Two pairs of opposite sides are equal. [3] Two pairs of opposite angles are equal. [4] One pair of opposite sides is parallel and equal in length. [5] The diagonals intersect at their respective midpoints.'

'In the quadrilateral ABCD inscribed in a circle, where AB = BC = 1, BD = √7, and DA = 2, find:\n1. The position of point A\n2. The length of side CD\n3. The area of quadrilateral ABCD S'

'Find the circumcenter O of triangle ABC.'

'Given a quadrilateral ABCD inscribed in circle PR with AB = 4, BC = 5, CD = 7, DA = 10, find the area S of quadrilateral ABCD.'

'Determine the minimum length of the diagonal in a rectangle with a length of 40 cm. Also, describe the shape of the rectangle at this minimum. Let the vertical length of the rectangle be x cm, then the horizontal length is (20-x) cm. Since x>0 and 20-x>0, we have 0<x<20. Denote the length of the diagonal as l cm, l^2 =x^2+(20-x)^2 =2 x^2-40 x+400 =2(x-10)^2+200 (1) where l^2 reaches the minimum value of 200 at x=10. Since l>0, when l^2 is minimized, l is also minimized. Thus, the minimum value of the diagonal length l is sqrt(200)=10 sqrt(2)(cm). At this point, the horizontal length is also 10 cm, making the rectangle a square.'

'(2) Consider cases based on the length of one side. 11) A square formed by two vertically adjacent lines and two horizontally adjacent lines.'

'Take point O on the plane and define two perpendicular lines at point O, as shown in the diagram to the right. These are called the x-axis and y-axis, respectively. Point O is called the origin. In this case, if point A is located at coordinates (3, 2), please provide its x-coordinate and y-coordinate.'

'In the figure to the right, label all the vertices of an equilateral triangle of side length 2 and the midpoints of each side with numbers 1 to 6, corresponding to the rolls of a die. Roll the die 3 times and connect the numbers obtained to form a geometric shape. Find the expected value of the area of the resulting shape.'

'There are 10 non-intersecting lines on a plane, with no three lines passing through the same point. If two of the 10 lines are parallel, determine the number of intersection points and triangles formed by these 10 lines.'

'Example 4: Parabolic Antenna\nA parabola is called a parabola in English. The surface of a parabolic antenna used for satellite broadcasting reception is in the shape of the surface that is formed by rotating a parabola around its axis.'

"Circles O and O' with radii 5 and 8, respectively, are externally tangent at point A. Let B and C be the points where the common external tangent of these two circles touches circles O and O'. Extend BA to intersect circle O' at point D.\n(2) Prove that points C, O', and D are collinear.\n(3) Find the ratios of AB:AC:BC."

'Calculate the number of parallelograms formed by 3 parallel lines and 5 lines intersecting them.'

'Distance between two points\n(1) The distance between two points A(x1, y1) and B(x2, y2) on the coordinate plane is\nAB=√((x2-x1)^2+(y2-y1)^2)\nIn particular, the distance between the origin O and point A(x1, y1) is OA=√(x1^2+y1^2)\n(2) The distance between two points A(x1, y1, z1) and B(x2, y2, z2) in coordinate space is\nAB=√((x2-x1)^2+(y2-y1)^2+(z2-z1)^2)\nIn particular, the distance between the origin O and point A(x1, y1, z1) is OA=√(x1^2+y1^2+z1^2)'

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'For a regular decagon, find the following:\n(1) The number of diagonals\n(2) The number of triangles with 3 of the vertices of the decagon as vertices\n(3) The number of triangles from (2) that share only one side with the decagon'

'Find the distance between A and P.'

'Prove that the centroids of triangles ABC and DEF coincide.'

'The minimum value is 10√2 cm, square'

'For the given line segment AB, draw the following points.'

'Mathematics I\nTherefore, the area of triangle (ABC) is\n\nEX 1 A piece of paper in the shape of an equilateral triangle with a side length of (10 cm) is given. Let the vertices of this equilateral triangle be (A, B, C), and let point (P) on side (BC) be at a distance of (2 cm) from point (B). When folding this equilateral triangle paper so that point (A) coincides with point (P), let the intersection points of the folds with sides (AB, AC) be (D, E) respectively. At this time, if (AD=) is A(cm), (AE=) is B(cm), and the area of △ADE is C(cm^{2}).\n[From Kyoto Makie University]'

'In triangle ABC, where BC=17, CA=10, AB=9. Find the value of sinA, the area of triangle ABC, the radius of the circumcircle, and the radius of the incircle.'

'Basic example 85 The length of the segment cut by the parabola from the x-axis\n(1) Find the length of the segment cut by the graph of the quadratic function y=-x^{2}+3x+3 from the x-axis.\n(2) Prove that the length of the segment cut by the graph of the quadratic function y=x^{2}-2ax+a^{2}-3 from the x-axis is constant regardless of the value of the constant a.'

'Given AB=2√2, BC=4√2, CA=2√6, find the lengths of the sides of the triangle.'

'(2) In triangle ABC, if BC = 5, CA = 3, AB = 7. Let D and E be the points where angle A and its exterior bisector intersect line BC, respectively. Find the length of segment DE.'

'On a plane, there are 10 lines such that no three of them intersect at the same point. When exactly 2 of the 10 lines are parallel, determine the number of intersection points formed by these 10 lines and the number of triangles formed.'

'Plot the following points:\n(1) Point P equidistant from sides AB, BC, CA\n(2) Point Q equidistant from points A, B, C'

'Draw the following circles based on the circle O and chord AB shown on the right. Note that points P and Q are different from A and B, and are not on the perpendicular bisector of chord AB.'

'For a right triangle with side lengths a, b, c, where the circumradius is 3/2 and the inradius is 1/2, answer the following questions. Assume a ≥ b ≥ c:'

'Find the minimum length of the diagonal in a rectangle with a length of 40 cm. Also, determine what kind of rectangle it will be at that time.'

'Write the following mathematical terms and their corresponding definitions in Japanese.'

'The length of the shorter side is at least 1 meter and at most 3 meters'

'Find the equation of the parabola obtained by symmetrically moving the parabola y=-2x^2+3x-5 with respect to the following lines or points.'

'Find the angle θ formed by the following two lines. Assume 0° ≤ θ ≤ 90°.(1) AB and FG (2) AE and BG (3) AF and CD'

'In a equilateral triangle ABC with side length 1, let D divide BC in the ratio 1:2, let E divide CA in the ratio 1:2, let F divide AB in the ratio 1:2. Let P be the intersection of BE and CF, Q be the intersection of CF and AD, and R be the intersection of AD and BE. Find the area of triangle PQR.'

'In triangle ABC, if AB = 6, BC = 7, CA = 5, find the radii of the circumcircle R and the incircle r.'

'Please provide the terms and the pages that have shapes circumscribed by two circles.'

'On the semicircle with a radius of 1, the point whose x-coordinate is 1/2 is the point P. The angle we are looking for is ∠AOP.'

'Find the coordinates of the point Q, which is symmetrical to the point P(3, 4) with respect to the line y = 2x + 1.'

'Let the lengths of the sides of triangle ABC be a, b, c. If (a+b) : (b+c) : (c+a) = 4 : 5 : 6 and the area is 15√3, then find the circumradius R and inradius r of triangle ABC.'

'Incenter of a triangle'

'In triangle ABC, with the radius of the circumcircle denoted as R. When A=30 degrees, B=105 degrees, and a=5, find R and c.'

'When symmetrically moved about the origin, the vertex is at the point \\( \\left(-\\frac{3}{4}, \\frac{31}{8}\\right) \\), forming 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 { also valid }\\right) \\]'

'Circle O intersects with circle B at points P and Q, and further intersects with the diameter FG of circle B at points A and center B. In addition, E is the intersection point of lines PQ and FG. If EA = x, AB = a, and BG = b, express x in terms of a and b.'

'Find the distance between the following two points.'

'There is an equilateral triangle paper with a side length of 10 cm. Let A, B, and C be the vertices of this equilateral triangle, and point P be a point on edge BC such that BP=2 cm. When folding this equilateral triangle paper so that vertex A coincides with point P, let the intersections of edges AB, AC, and the fold be denoted as D and E, respectively. At this point, AD= cm, AE= cm, and the area of triangle ADE is cm².'

'For a triangle ABC that is not an equilateral triangle, with circumcenter O, centroid G, and orthocenter H, prove the following: (1) Let L be the midpoint of side BC, and M, N be the midpoints of segments GH and AG respectively. Prove that quadrilateral OLMN is a parallelogram. You may use the fact that AH=2OL. (2) Prove that point G lies on segment OH. (3) Prove that OG:GH=1:2.'

'Find the length of the line segment cut from the x-axis by the graph of the following two quadratic functions:\n(A) y = 2x^2 - 8x - 15\n(B) y = x^2 - (2a + 1)x + a(a + 1) (where a is a constant)'

'Positional relationship between a parabola and the x-axis'

'Summary of Determining the Shape of a Triangle'

"There are two circles P and Q that intersect with two circles O and O'. As shown in the figure on the right, draw a line from point A, which is beyond P of segment QP, tangent to circle O and intersecting with circle O', with the points of tangency being C, and the points of intersection being B and D. If AB=a, BC=b, CD=c, express c in terms of a and b."

"Prove the following using the converse of Ceva's theorem:\n1. The three medians of a triangle intersect at one point.\n2. The three angle bisectors of a triangle intersect at one point."

'67 diagrams; in order of vertices and axes: (1) point (2, -1), line x=2 (2) point (-2, -3), line x=-2 (3) point (1, 1), line x=1'

'Quadrilateral ABCD is inscribed in a circle O, with AB=3, BC=CD=√3, and cos ∠ABC=√3/6. Find: (1) The length of segment AC (2) The length of side AD (3) The radius R of circle O'

'Find the area of the following shapes.\n1. Parallelogram ABCD with AB=2, BC=3, and ∠ABC=60 degrees\n2. Regular octagon circumscribed around a circle with radius 10'

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'In triangle ABC, when a=13, b=7, and c=15, find A.'

'For a right triangle where the sum of the lengths of the two legs is 16, what shape maximizes the area? Also, calculate the maximum value.'

"There are three cases for the position relationship between a circle and a line. Here, r is the radius of the circle, and d is the distance between the center of the circle and the line. [1] Intersection at 2 points (2 shared points) 0 ≤ d < r [2] Tangent (1 shared point) 0 ≤ d < r [3] Disjoint (no shared points) 0 ≤ d < r When there is only one shared point, the circle and the line are tangent, and this line is called the tangent, with the shared point being the point of tangency. Let's first investigate the properties of the tangents of a circle."

'Determine the conditions for the triangles to exist'

'In triangle ABC, where AB=6, BC=a, and CA=4, let M and N be the midpoints of BC and CA, respectively. (1) Find the value of a when AM=√10. (2) When a is the value from (1), find the length of segment BN.'

'(2) The longest side is CA, so AB + BC = 18, CA < AB + BC, therefore triangle ABC exists.'

"Let's review Example 65! Let's use the fact that the lengths of the two tangents drawn from a point outside the circle are equal. Once again, let's try eliminating unnecessary shapes."

'TRAINING 112 (1)\nLet the point O on the flat square be the origin, consider the coordinate plane with the east direction as the positive direction of the x-axis and the north direction as the positive direction of the y-axis.\nPoint A is located 28 units east of point O. Additionally, point P is south of the line connecting points O and A.\nPoint P is at a distance of 25 from O and 17 from A.\n(1) Find the coordinates of point A.\n(2) Find the coordinates of point P.'

'Practice 3: Incenter, Circumcenter, Centroid of a Triangle'

'Solve the following problems for triangle ABC:\n(1) Find the value of cos A.\n(2) Find the area S of triangle ABC.\n(3) Find the radius r of the incircle of triangle ABC.'

'Calculate the values of trigonometric functions in a right triangle'

'Proof that there are four points on a circle'

'One of the intersection points of two graphs is the point (-1,0)'

'Prove that the points B, C, F, E lie on a single circle when a perpendicular line AD is drawn from vertex A of acute triangle ABC to side BC, and perpendicular lines DE, DF are drawn from D to sides AB and AC, respectively.'

'In right triangle ABC, with AB > AC and ∠A = 90°, draw the perpendicular AD from vertex A to edge BC.'

'In the pentagon ABCDE circumscribed about a circle, where AB = 7, BC = 3, CD = 5, DE = 6, ∠BCD = 120° and ∠A = 82°, find:\n(1) The length of segment BD\n(2) The length of segment AD\n(3) The length of side AE\n(4) The area of quadrilateral ABDE'

'When A=90^{\\circ}, \\sin A=\\sin 90^{\\circ}=1, so 2R \\sin A=2R \\cdot 1=2R. Also, side BC is the diameter of the circumcircle of triangle ABC, so a=2R, hence a=2R \\sin A.'

'(1) \\\\( \\theta=30^{\\circ}, \\\\ 150^{\\circ} \\\\\\\n(2) \\\\( \\theta=45^{\\circ} \\\\\\\n(3) \\\\( \\theta=120^{\\circ} \\\\\\\n'

'66 diagram; (1) point (-1,0) and line x=-1 in order of vertices and axis, (2) point (1,1) and line x=1'

'Chapter 7 Applications to Triangles'

'In △ABC, suppose AB = 7√3 and ∠ACB = 60°. What is the radius of the circumcircle O of △ABC? Let point P move on the arc AB containing point C of the circumcircle O.'

'(1) Find the measures of the three angles of triangle ABC where ∠A=90°, AB=2, and BC=3.\n(2) Find the lengths of the three sides of triangle ABC where ∠A=70° and ∠B=∠C.'

'(1) ∠C < ∠B < ∠A\n(2) CA = AB < BC'

'In the diagram on the right, point I is the incenter of triangle ABC. Find the following: (1) 𝛼 (2) AI:ID'

'Consider a coordinate plane with point O as the origin, eastward as the positive direction of the x-axis, and northward as the positive direction of the y-axis.'

'The graph of the quadratic function y = ax^2 + 2ax + a + 6 (a≠0) intersects the x-axis at two points P and Q, and the length of line segment PQ is 2√6. Determine the value of the constant a.'

'(1) 60°\n(2) 50°\n(3) 140°'

'In the figure on the right, point O is the circumcenter of triangle ABC. Find the angles alpha and beta.'

"When thinking as described in the question, it is unclear where to apply the properties of the learned shapes. First, point A is the point of tangency of the two circles O and O', so let's draw the common tangents of the two circles passing through point A. By focusing on a part of the figure, the relevant properties will become apparent."