2009 IMO Problems/Problem 2

Problem

Let $ABC$ be a triangle with circumcentre $O$ . The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$ respectively. Let $K,L$ and $M$ be the midpoints of the segments $BP,CQ$ and $PQ$ , respectively, and let $\Gamma$ be the circle passing through $K,L$ and $M$ . Suppose that the line $PQ$ is tangent to the circle $\Gamma$ . Prove that $OP=OQ$ .

Author: Sergei Berlov, Russia

Solution

Diagram

$[asy] dot("O", (50, 38), NW); dot("A", (40, 100), N); dot("B", (0, 0), S); dot("C", (100, 0), S); dot("Q", (24, 60), W); dot("P", (52, 80), E); dot("L", (62, 30), SE); dot("M", (38, 70), N); dot("K", (27, 42), W); draw((100, 0)--(24, 60), dotted); draw((0, 0)--(52, 80), dashed); draw((0, 0)--(100, 0)--(40, 100)--cycle); draw((24, 60)--(52, 80)); draw((27, 42)--(38, 70)--(62, 30)--cycle); draw(circle((49, 49), 23)); label("$\Gamma$", (72, 49), E); draw(circle((50, 38), 63)); label("$\omega$", (-13, 38), NW); [/asy]$ Diagram by qwertysri987

By parallel lines and the tangency condition, $\angle APM\cong \angle LMP \cong \angle LKM.$ Similarly, $\angle AQP\cong \angle KLM,$ so AA similarity implies $\triangle APQ\sim \triangle MKL.$ Let $\omega$ denote the circumcircle of $\triangle ABC,$ and $R$ its circumradius. As both $P$ and $Q$ are inside $\omega,$

$\begin{align*} R^2-QO^2&=\text{Pow}_{\omega}(Q)\\ &=QB\cdot AQ \\ &=2AQ\cdot MK\\ &=2AP\cdot ML\\ &=AP\cdot PC\\ &=\text{Pow}_{\omega}(P)\\ &=R^2-PO^2. \end{align*}$ It follows that $OP=OQ.$ $\blacksquare$

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2009 IMO Problems/Problem 2

Problem

Solution

Diagram