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Concurrent lines

Source: Romania TST 2 2010, Problem 3

August 25, 2012
geometrycircumcirclepower of a pointradical axisgeometry proposed

Problem Statement

Let γ1\gamma_1 and γ2\gamma_2 be two circles tangent at point TT, and let ℓ1\ell_1 and ℓ2\ell_2 be two lines through TT. The lines ℓ1\ell_1 and ℓ2\ell_2 meet again γ1\gamma_1 at points AA and BB, respectively, and γ2\gamma_2 at points A1A_1 and B1B_1, respectively. Let further XX be a point in the complement of γ1∪γ2∪ℓ1∪ℓ2\gamma_1 \cup \gamma_2 \cup \ell_1 \cup \ell_2. The circles ATXATX and BTXBTX meet again γ2\gamma_2 at points A2A_2 and B2B_2, respectively. Prove that the lines TXTX, A1B2A_1B_2 and A2B1A_2B_1 are concurrent.
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