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Geo with two circles and a variable line

Source: Czech-Polish-Slovak Match 2022 P3

September 3, 2022
geometry

Problem Statement

Circles Ω1\Omega_1 and Ω2\Omega_2 with different radii intersect at two points, denote one of them by PP. A variable line ll passing through PP intersects the arc of Ω1\Omega_1 which is outside of Ω2\Omega_2 at X1X_1, and the arc of Ω2\Omega_2 which is outside of Ω1\Omega_1 at X2X_2. Let RR be the point on segment X1X2X_1X_2 such that X1P=RX2X_1P = RX_2. The tangent to Ω1\Omega_1 through X1X_1 meets the tangent to Ω2\Omega_2 through X2X_2 at TT. Prove that line RTRT/is tangent to a fixed circle, independent of the choice of ll.
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