IMO Shortlist 2008 problem G7

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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} intersects on k.

Author: Vladimir Shmarov, Russia
Izvor: Međunarodna matematička olimpijada, shortlist 2008