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common chord of 2 circles through A 2014 Cuba 2.3

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September 20, 2024
geometrycircles

Problem Statement

Let Γ1\Gamma_1 and Γ2\Gamma_2 be the circles with diameters APAP and AQAQ. Let TT be another point of intersection of the circles Γ1\Gamma_1 and Γ2\Gamma_2. Let Q1Q_1 be another point of intersection of the circle Γ1\Gamma_1 and the line AQAQ, and P1P_1 the other point of intersection of the circle Γ2\Gamma_2 and the line APAP. The circle Γ3\Gamma_3 passes through the points TT, PP and P1P_1 and the circle Γ4\Gamma_4 passes through the points TT, QQ and Q1Q_1. Prove that the line containing the common chord of the circles Γ3\Gamma_3 and Γ4\Gamma_4 passes throughAA.
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