IMO 2003 Problem 4
IMO 2003 Problem 4 is a plane geometry problem from the International Mathematical Olympiad 2003, Problem 4: feet of the perpendiculars from a vertex of a cyclic quadrilateral.
Problem
Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$ and $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$ and $AB$ respectively. Show that $PQ = QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ meet on $AC$.
Source International Mathematical Olympiad 2003, Problem 4
The diagram
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