Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be a triangle with bisectors
![AA_1,BB_1, CC_1](/media/m/1/f/d/1fdb1246f5bfa3c9ea1e631bfea38897.png)
(
![A_1 \in BC](/media/m/5/1/5/5157339b2263f41eb36a13113b2423f7.png)
, etc.) and
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
their common point. Consider the triangles
![MB_1A, MC_1A,MC_1B,MA_1B,MA_1C,MB_1C](/media/m/6/5/d/65d71693667d6b46fdaeead3db3fe006.png)
, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then
%V0
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 $AB = BC = CA.$