Question 30
Let $\triangle ABC$ be an acute-angled triangle. Let $D$ be a point on the segment $BC$, $I$ the incentre of $\triangle ABC$. The circumcircle of $\triangle ABD$ meets $BI$ at $P$ and the circumcircle of $\triangle ACD$ meets $CI$ at $Q$. If the area of $\triangle PID$ and the area of $\triangle QID$ are equal, prove that $PI \times QD = QI \times PD$.