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collinear midpoints on fixed line - All-Russian MO 2004 Regional (R4) 10.7

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September 27, 2024
geometrycollinearLocuscircles

Problem Statement

Circles ω1\omega_1 and ω2\omega_2 intersect at points AA and BB. At point AA to ω1\omega_1 and ω2\omega_2 the tangents ℓ1\ell_1 and ℓ2\ell_2 are drawn respectively. The points T1T_1 and T2T_2 are chosen respectively on the circles ω1\omega_1 and ω2\omega_2 so that the angular measures of the arcs T1AT_1A and AT2AT_2 are equal (the measure of the circular arc is calculated clockwise). The tangent t1t_1 at the point T1 T_1 to the circle ω1\omega_1 intersects ℓ2\ell_2 at the point M1M_1. Similarly, the tangent t2t_2 at the point T2T_2 to the circle ω2\omega_2 intersects ℓ1\ell_1 at point M2M_2. Prove that the midpoints of the segments M1M2M_1M_2 are on the same a straight line that does not depend on the position of points T1T_1, T2T_2.
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