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fixed point's revenge, 4 circles related, tangent circles related

Source: Ukraine TST 2009 p8

May 2, 2020
geometryFixed pointcirclestangent circlesfixed

Problem Statement

Two circles γ1,γ2\gamma_1, \gamma_2 are given, with centers at points O1,O2O_1, O_2 respectively. Select a point KK on circle γ2\gamma_2 and construct two circles, one γ3\gamma_3 that touches circle γ2\gamma_2 at point KK and circle γ1\gamma_1 at a point AA, and the other γ4\gamma_4 that touches circle γ2\gamma_2 at point KK and circle γ1\gamma_1 at a point BB. Prove that, regardless of the choice of point K on circle γ2\gamma_2, all lines ABAB pass through a fixed point of the plane.