Question 93
Let the incircle of a scalene triangle $ABC$ touch the sides $BC$, $CA$, $AB$ at the points $A'$, $B'$, $C'$, respectively. Let $D$ be the midpoint of the arc $B'C'$ that does not contain $A'$ ($D$ lies on the opposite side of $A'$). If the common tangent of the incircles of triangles $A'CB'$ and $A'C'B$ (opposite to the common tangent $BC$ with respect to the line joining their centers) intersects the lines $A'D$ and $A'C'$ at the points $E$ and $F$ respectively, show that $C'EF$ is an isosceles triangle.