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Problem #PRU-110157

Problems Geometry Plane geometry Plane transformations Similarity Homothetic transformations, rotations of homothetic transformations Circles Connection between the arc length, the angle, and the chord Calculus Derivative Derivative and multiple roots

14–16 5.0

The circles $σ_{1}$ and $σ_{2}$ intersect at points $A$ and $B$ . At the point $A$ to $σ_{1}$ and $σ_{2}$ , respectively, the tangents $l_{1}$ and $l_{2}$ are drawn. The points $T_{1}$ and $T_{2}$ are chosen respectively on the circles $σ_{1}$ and $σ_{2}$ so that the angular measures of the arcs $T_{1} A$ and $A T_{2}$ are equal (the arc value of the circle is considered in the clockwise direction). The tangent $t_{1}$ at the point $T_{1}$ to the circle $σ_{1}$ intersects $l_{2}$ at the point $M_{1}$ . Similarly, the tangent $t_{2}$ at the point $T_{2}$ to the circle $σ_{2}$ intersects $l_{1}$ at the point $M_{2}$ . Prove that the midpoints of the segments $M_{1} M_{2}$ are on the same line, independent of the positions of the points $T_{1}, T_{2}$ .

Problem #PRU-57128

Problems Geometry Plane geometry Plane transformations Similarity Homothetic transformations, rotations of homothetic transformations Locus

12–13 2.0

The point $A$ is fixed on a circle. Find the locus of the point $X$ which divides the chords that end at point $A$ in a $1 : 2$ ratio, starting from the point $A$ .

Problem #PRU-57807

Problems Geometry Plane geometry Plane transformations Parallel translation

13–14 2.0

Prove that, when a circle is translated it becomes a circle.

Problem #PRU-57808

Problems Geometry Plane geometry Plane transformations Parallel translation

13–15 4.0

Two circles of radius $R$ touch at point $E$ . On one of them, point $B$ is chosen and on the other point $D$ is chosen. These points have a property of $∠ B E D = 90^{\circ}$ . Prove that $B D = 2 R$ .

Problem #PRU-57809

Problems Geometry Plane geometry Plane transformations Parallel translation

13–15 3.0

Two circles of radius $R$ intersect at points $D$ and $B$ . Let $F$ and $G$ be the points of intersection of the middle perpendicular to the segment $B D$ with these circles lying on one side of the line $B D$ . Prove that $B D^{2} + F G^{2} = 4 R^{2}$ .

Problem #PRU-57810

Problems Geometry Plane geometry Plane transformations Parallel translation

13–14 2.0

Inside the rectangle $A B C D$ , the point $E$ is taken. Prove that there exists a convex quadrilateral with perpendicular diagonals of lengths $A B$ and $B C$ whose sides are equal to $A E$ , $B E$ , $C E$ , $D E$ .

Problem #PRU-57914

Problems Geometry Plane geometry Plane transformations Isometry of the plane Rotations

12–14 2.0

Prove that a circle transforms into a circle when it is rotated.

Problem #PRU-57915

Problems Geometry Plane geometry Plane transformations Isometry of the plane Rotations

13–15 3.0

Prove that a convex $n$ -gon is regular if and only if it is transformed into itself when it is rotated through an angle of $360^{\circ} / n$ with respect to some point.

Problem #PRU-57916

Problems Geometry Plane geometry Plane transformations

13–14 2.0

Prove that the triangle $A B C$ is regular if and only if, by turning it by $60^{\circ}$ (either clockwise or anticlockwise) with respect to point A, its vertex B moves to $C$ .

Problem #PRU-57918

Problems Geometry Plane geometry Plane transformations Isometry of the plane Rotations

13–16 2.0

Two perpendicular straight lines are drawn through the centre of the square. Prove that their intersection points with the sides of a square form a square.