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2 tangents on different circles concurrent with 1 common tangent

Source: 2022 Dutch IMO TST 1.2

December 3, 2022
geometryconcurrencycommon tangents

Problem Statement

Two circles Γ1\Gamma_1 and Γ2\Gamma_2are given with centres O1O_1 and O2O_2 and common exterior tangents ℓ1\ell_1 and ℓ2\ell_2. The line ℓ1\ell_1 intersects Γ1\Gamma_1 in AA and Γ2\Gamma_2 in BB. Let XX be a point on segment O1O2O_1O_2, not lying on Γ1\Gamma_1 or Γ2\Gamma_2. The segment AXAX intersects Γ1\Gamma_1 in Y≠AY \ne A and the segment BXBX intersects Γ2\Gamma_2 in Z≠BZ \ne B. Prove that the line through YY tangent to Γ1\Gamma_1 and the line through ZZ tangent to Γ2\Gamma_2 intersect each other on ℓ2\ell_2.
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