Question 28
Let $D$ be a point on the side $AC$ of a triangle $ABC$. Let $E$ and $F$ be the points symmetric to $D$ with respect to the angle bisectors of $\angle A$ and $\angle C$, respectively. Prove that the midpoint of the segment $EF$ lies on the line $A_0C_0$, where $A_0$ and $C_0$ are the points of tangency of the incircle with the sides $BC$ and $AB$, respectively. (T. Emelyanova)