MathNet 2015-part-04 — 191 problems (EN–VI) Question 124

Let $I$ be the center of the inscribed circle of triangle $ABC$, with $AB \neq AC$. Let $M$ be the midpoint of side $BC$, and $D$ the projection of $I$ on $BC$. The circle with center $M$ and radius $MD$ intersects line $AI$ at $P$ and $Q$. Show that $\angle BAC + \angle PMQ = 180^\circ$.

Laurențiu Ploscaru