Problem

Source: Paraguayan Mathematical Olympiad 2013

Tags: geometry



Let $ABC$ be an obtuse triangle, with $AB$ being the largest side. Draw the angle bisector of $\measuredangle BAC$. Then, draw the perpendiculars to this angle bisector from vertices $B$ and $C$, and call their feet $P$ and $Q$, respectively. $D$ is the point in the line $BC$ such that $AD \perp AP$. Prove that the lines $AD$, $BQ$ and $PC$ are concurrent.