Question 1
(BUL 1) Let $ABC$ be a triangle with bisectors $AA_1,BB_1,CC_1$ ($A_1\in BC$, etc.) and $M$ their common point. Consider the triangles $MB_1A,MC_1A,MC_1B,MA_1B,MA_1C,MB_1C$, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then $AB=BC=CA$.