IMO 2015 Problem 3
IMO 2015 Problem 3 is a plane geometry problem from the International Mathematical Olympiad 2015, Problem 3: tangent circumcircles in an acute triangle.
Problem
Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma$ be its circumcircle, $H$ its orthocentre, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^\circ$, and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^\circ$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different, and lie on $\Gamma$ in this order. Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.
Source International Mathematical Olympiad 2015, Problem 3
The diagram
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