Question 72
Let $ABC$ be a triangle with $\angle ABC \neq \angle BCA$. The incircle $k$ of triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$ respectively. The segment $AD$ meets $k$ a second time at $P$. Let $Q$ be the intersection point of $EF$ with the perpendicular to $AD$ through $P$. Let $X$ and $Y$ be the intersection points of $AQ$ with $DE$ and with $DF$ respectively. Show that $A$ is the midpoint of $XY$.