Question 98
Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Let $A_{1}$ be the midpoint of the arc $\widehat{BC}$ not containing $A$; $B_{1}$ the midpoint of the arc $\overparen{CA}$ not containing $B$; $C_{1}$ the midpoint of the arc $\overparen{AB}$ not containing $C$. Finally, let $A_{2}$ be the point for which $AB_{1}A_{2}C_{1}$ is a parallelogram; $B_{2}$ the point for which $BC_{1}B_{2}A_{1}$ is a parallelogram; $C_{2}$ the point for which $CA_{1}C_{2}B_{1}$ is a parallelogram.
Prove that the circumcircles of $ABC$ and of $A_{2}B_{2}C_{2}$ are concentric.
Prove that the circumcircles of $ABC$ and of $A_{2}B_{2}C_{2}$ are concentric.