Question 161
Let $ABC$ be a triangle, which is not an isosceles triangle. Let $\Gamma$ be its circum-circle and $I$ be its in-center. Let $D$ and $E$ be the points of tangency of the in-circle of $ABC$ to the side $AB$ and $AC$, respectively. Let $P$ be the point of intersection, different from $B$, of $\Gamma$ and the circum-circle of triangle $BEI$, and $Q$ be the point of intersection, different from $C$, of $\Gamma$ and the circum-circle of triangle $CDI$. Show that four points $D$, $E$, $P$, $Q$ are con-cyclic.