Question 4
Let an acute triangle $ABC$ be inscribed in a circle with center $O$, with $BC\gt AB\gt AC$. Line $OA$ meets line $BC$ at $A_1$, and line $OB$ meets line $CA$ at $B_2$. Let $B_1,C_1,C_2,A_2$ be the centers of circles $(AA_1B),(AA_1C),(BB_2C),(BB_2A)$, respectively. Prove that:
1. Triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.
2. These triangles are congruent if and only if angle $C$ of triangle $ABC$ equals $60^\circ$.
Here $(XYZ)$ denotes the circle through $X,Y,Z$.
1. Triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.
2. These triangles are congruent if and only if angle $C$ of triangle $ABC$ equals $60^\circ$.
Here $(XYZ)$ denotes the circle through $X,Y,Z$.