Question 107
The inscribed circle in $\triangle ABC$ ($AC \neq BC$) is tangent to its sides $AB$, $BC$, and $CA$ at points $D$, $E$, and $F$, respectively. Let $P$ be the foot of the perpendicular from $D$ to $EF$ ($P \in EF$). If the circles circumscribed about $\triangle ABC$ and $\triangle EFC$ intersect for the second time at point $Q$, prove that $\angle PQC = 90^\circ$.
(Stoyan Boev)
(Stoyan Boev)