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Three murderous tangent circles

Source: Own. Malaysian IMO TST 2024 P6

April 21, 2024
geometry

Problem Statement

Let ω1\omega_1, ω2\omega_2, ω3\omega_3 are three externally tangent circles, with ω1\omega_1 and ω2\omega_2 tangent at AA. Choose points BB and CC on ω1\omega_1 so that lines ABAB and ACAC are tangent to ω3\omega_3. Suppose the line BCBC intersect ω3\omega_3 at two distinct points, and XX is the intersection further away to BB and CC than the other one.
Prove that one of the tangent lines of ω2\omega_2 passing through XX, is also tangent to an excircle of triangle ABCABC.
Proposed by Ivan Chan Kai Chin
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