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Problems Geometry Plane geometry Triangles Similar triangles Circles Tangent lines and tangent circles Tangent lines to circles

15–16 4.0

Three circles are constructed on a triangle, with the medians of the triangle forming the diameters of the circles. It is known that each pair of circles intersects. Let $C_{1}$ be the point of intersection, further from the vertex $C$ , of the circles constructed from the medians $A M_{1}$ and $B M_{2}$ . Points $A_{1}$ and $B_{1}$ are defined similarly. Prove that the lines $A A_{1}$ , $B B_{1}$ and $C C_{1}$ intersect at the same point.

Problem #PRU-56684

Problems Geometry Plane geometry Circles Tangent lines and tangent circles Tangent lines to circles

13–14 2.0

From a point $A$ the tangents $A B$ and $A C$ are drawn to a circle with center $O$ . Prove that if from the point $M$ the segment $A O$ is visible at an angle of $90^{\circ}$ , then the segments $O B$ and $O C$ are also visible from it at equal angles.