Question 157
Let $ABC$ be an acute scalene triangle. The incircle of $ABC$ touches $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Let $X$, $Y$, $Z$ be feet of the altitudes from $A$, $B$, $C$ to the sides $BC$, $CA$, $AB$ respectively. Let $A'$, $B'$, $C'$ be the reflections of $X$, $Y$, $Z$ in $EF$, $FD$, $DE$ respectively. Prove that triangles $ABC$ and $A'B'C'$ are similar.