We solve problems

Problems Geometry Plane geometry Circles Inscribed angle Angles supported by congruent arcs and chords

The vertex $A$ of the acute-angled triangle $A B C$ is connected by a segment with the center $O$ of the circumscribed circle. The height $A H$ is drawn from the vertex $A$ . Prove that $∠ B A H = ∠ O A C$ .

Problem #PRU-56541

Problems Geometry Plane geometry Circles Inscribed angle Angles supported by congruent arcs and chords

12–13 2.0

The vertex $A$ of the acute-angled triangle $A B C$ is connected by a segment with the center $O$ of the circumscribed circle. The height $A H$ is drawn from the vertex $A$ . Prove that $∠ B A H = ∠ O A C$ .

Problem #PRU-56543

Problems Geometry Plane geometry Circles Inscribed angle Angles supported by congruent arcs and chords

12–13 2.0

From an arbitrary point $M$ lying within a given angle with vertex $A$ , the perpendiculars $M P$ and $M Q$ are dropped to the sides of the angle. From point $A$ , the perpendicular $A K$ is dropped to the segment $P Q$ . Prove that $∠ P A K = ∠ M A Q$ .

Problem #PRU-56555

Problems Geometry Plane geometry Circles Inscribed angle An angle between two chords or two secants

13–13 2.0

On a circle, the points $A, B, C, D$ are given in the indicated order. $M$ is the midpoint of the arc $A B$ . We denote the intersection points of the chords $M C$ and $M D$ with the chord $A B$ by $E$ and $K$ . Prove that $K E C D$ is an inscribed quadrilateral.

Problem #PRU-56562

Problems Geometry Plane geometry Circles Inscribed angle An angle between a secant and a chord

13–13 2.0

Two circles intersect at the points $P$ and $Q$ . Through the point $A$ of the first circle, the lines $A P$ and $A Q$ are drawn intersecting the second circle at points $B$ and $C$ . Prove that the tangent at point $A$ to the first circle is parallel to the line $B C$ .

Problem #PRU-56572

Problems Geometry Plane geometry Circles Connection between the arc length, the angle, and the chord

13–14 2.0

The isosceles trapeziums $A B C D$ and $A_{1} B_{1} C_{1} D_{1}$ with corresponding parallel sides are inscribed in a circle. Prove that $A C = A_{1} C_{1}$ .

Problem #PRU-56573

Problems Geometry Plane geometry Circles Connection between the arc length, the angle, and the chord

13–14 2.0

From the point $M$ , moving along a circle the perpendiculars $M P$ and $M Q$ are dropped onto the diameters $A B$ and $C D$ . Prove that the length of the segment $P Q$ does not depend on the position of the point $M .$

Problem #PRU-56581

Problems Geometry Plane geometry Circles Four points on a circle

13–14 2.0

From an arbitrary point $M$ on the side $B C$ of the right angled triangle $A B C$ , the perpendicular $M N$ is dropped onto the hypotenuse $A B$ . Prove that $∠ M A N = ∠ M C N$ .

Problem #PRU-56582

Problems Geometry Plane geometry Circles Four points on a circle

13–14 2.0

The diagonals of the trapezium $A B C D$ with the bases $A D$ and $B C$ intersect at the point $O$ ; the points $B^{'}$ and $C^{'}$ are symmetrical to the vertices $B$ and $C$ with respect to the bisector of the angle $B O C$ . Prove that $∠ C^{'} A C = ∠ B^{'} D B$ .

Problem #PRU-56593

Problems Geometry Plane geometry Circles Inscribed angle Angles supported by congruent arcs and chords

13–14 2.0

On the circle, the points $A, B, C$ and $D$ are given. The lines $A B$ and $C D$ intersect at the point $M$ . Prove that $A C \times A D / A M = B C \times B D / B M$ .