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AB=CD, 2 equal circles and 2 tangent circles and a line (both tangent + secant)

Source: Belarus TST 2009 1.1

June 13, 2020
circlesgeometrytangent circlesequal segmentstangent

Problem Statement

Two equal circles S1S_1 and S2S_2 meet at two different points. The line ℓ\ell intersects S1S_1 at points A,CA,C and S2S_2 at points B,DB,D respectively (the order on ℓ\ell: A,B,C,DA,B,C,D) . Define circles Γ1\Gamma_1 and Γ2\Gamma_2 as follows: both Γ1\Gamma_1 and Γ2\Gamma_2 touch S1S_1 internally and S2S_2 externally, both Γ1\Gamma_1 and Γ2\Gamma_2 line ℓ\ell, Γ1\Gamma_1 and Γ2\Gamma_2 lie in the different halfplanes relatively to line ℓ\ell. Suppose that Γ1\Gamma_1 and Γ2\Gamma_2 touch each other. Prove that AB=CDAB=CD.
I. Voronovich
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