Problems

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.

Two circles touch at a point $A$. A common (outer) tangent touching the circles at points $C$ and $B$ is drawn. Prove that $\mathrm{\angle }CAB={90}^{\circ }$.

Two circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ touch at the point $A$. A straight line intersects $S_1$ at $A_1$ and $S_2$ at the point $A_2$. Prove that $O_1{A}_1\parallel O_2{A}_2$.

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