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2009 IMO Problems/Problem 2

Revision as of 06:17, 23 July 2009 by Bugi (talk | contribs) (Created page with '== Problem == Let <math>ABC</math> be a triangle with circumcentre <math>O</math>. The points <math>P</math> and <math>Q</math> are interior points of the sides <math>CA</math> …')
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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

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