Question 81
Let $\triangle ABC$ be a triangle with $\angle B \neq \angle C$. The incircle $I$ of a triangle $ABC$ touches the sides $BC, CA, AB$ at the points $D, E, F$, respectively. Let $P$ be the intersection of $AD$ and the incircle $I$, which is different from $D$.
Let $Q$ be the intersection of the line $EF$ and the line passing $P$ and perpendicular to $AD$, and let $X, Y$ be intersections of the line $AQ$ and $DE, DF$, respectively. Show that the point $A$ is the midpoint of $XY$.
Let $Q$ be the intersection of the line $EF$ and the line passing $P$ and perpendicular to $AD$, and let $X, Y$ be intersections of the line $AQ$ and $DE, DF$, respectively. Show that the point $A$ is the midpoint of $XY$.