IMO 2008 Problem 6
IMO 2008 Problem 6 is a plane geometry problem from the International Mathematical Olympiad 2008, Problem 6: A circle tangent to four lines of a quadrilateral.
Problem
Let $ABCD$ be a convex quadrilateral with $BA$ different from $BC$. Denote the incircles of triangles $ABC$ and $ADC$ by $k_1$ and $k_2$ respectively. Suppose that there exists a circle $k$ tangent to ray $BA$ beyond $A$ and to the ray $BC$ beyond $C$, which is also tangent to the lines $AD$ and $CD$. Prove that the common external tangents to $k_1$ and $k_2$ intersect on $k$.
Source International Mathematical Olympiad 2008, Problem 6
The diagram
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