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4 circles and non-trivial concurrency through the vertices

Source: All-Russian Olympiad 1993-1994,Final Round,10.7

September 6, 2008
geometryincentergeometric transformationhomothetyprojective geometrygeometry proposed

Problem Statement

Let Γ1,Γ2 \Gamma_1,\Gamma_2 and Γ3 \Gamma_3 be three non-intersecting circles,which are tangent to the circle Γ \Gamma at points A1,B1,C1 A_1,B_1,C_1,respectively.Suppose that common tangent lines to (Γ2,Γ3) (\Gamma_2,\Gamma_3),(Γ1,Γ3) (\Gamma_1,\Gamma_3),(Γ2,Γ1) (\Gamma_2,\Gamma_1) intersect in points A,B,C A,B,C. Prove that lines AA1,BB1,CC1 AA_1,BB_1,CC_1 are concurrent.
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