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Find the locus of intersection of common internal tangents

Source: XVII Sharygin Correspondence P6

March 2, 2021
geometrytangent circles

Problem Statement

Three circles Γ1,Γ2,Γ3\Gamma_1,\Gamma_2,\Gamma_3 are inscribed into an angle(the radius of Γ1\Gamma_1 is the minimal, and the radius of Γ3\Gamma_3 is the maximal) in such a way that Γ2\Gamma_2 touches Γ1\Gamma_1 and Γ3\Gamma_3 at points AA and BB respectively. Let ℓ\ell be a tangent to AA to Γ1\Gamma_1. Consider circles ω\omega touching Γ1\Gamma_1 and ℓ\ell. Find the locus of meeting points of common internal tangents to ω\omega and Γ3\Gamma_3.
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