Question 183
The inscribed circle of a triangle $ABC$ touches the sides $BC$, $CA$, $AB$ at $D$, $E$, and $F$ respectively. Let $X$, $Y$, and $Z$ be the incenters of triangles $AEF$, $BFD$, and $CDE$, respectively. Prove that $DX$, $EY$, and $CZ$ meet at one point.