Question 46
Given a triangle $ABC$, its $A$-excircle is the point where the external bisectors of angles $\angle B$ and $\angle C$ meet. Let $I_A$, $I_B$ and $I_C$ be the $A$, $B$, $C$-excenters of scalene triangle $ABC$, respectively, and $X$, $Y$ and $Z$ be the midpoints of $I_B I_C$, $I_C I_A$ and $I_A I_B$, respectively. The incircle of triangle $ABC$ touches sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$, respectively. Prove that lines $DX$, $EY$ and $FZ$ meet at a single point on line $IO$ that connects the incenter $I$ and the circumcenter $O$ of triangle $ABC$.