IMO 2009 Problem 2
IMO 2009 Problem 2 is a plane geometry problem from the International Mathematical Olympiad 2009, Problem 2: A circle through three midpoints tangent to PQ.
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 $OP = OQ$.
Source International Mathematical Olympiad 2009, Problem 2
The diagram
Drag any point and the whole construction follows; ▶ runs a point along its segment and back. Tap a point to pin it, double-click to go back to the start. Effects redraws the figure in grayscale for print, in colour, or with the givens in focus.