Problem

Source: 2018 Chile National Olympiad level 2 p6

Tags: geometry, collinear, orthocenter



Consider an acute triangle $ABC$ and its altitudes from $A$ ,$B$ that intersect the respective sides at $D ,E$. Let us call the point of intersection of the altitudes $H$. Construct the circle with center $H$ and radius $HE$. From $C$ draw a tangent line to the circle at point $P$. With center $B$ and radius $BE$ draw another circle and from $C$ another tangent line is drawn to this circle in the point $Q$. Prove that the points $D, P$, and $Q$ are collinear.