Question 21
Let $I$ be the incenter of triangle $ABC$, and let $D$ be the perpendicular projection of $I$ onto $BC$. Let $D'$ be the reflection of $D$ in $I$, and suppose $AD'=ID'$. The circle $\Gamma$ with center $D'$ passes through $A,I$ and meets $AB,AC$ again at $X,Y$, respectively. Let $Z$ be the point of $\Gamma$ for which $AZ\perp BC$. Prove that $AD,D'Z,XY$ are concurrent.