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Congruent triangles with tangent intersections

Source: 2023 Israel National Olympiad P3

December 16, 2022
geometrycongruent trianglesgeometric transformationreflection

Problem Statement

A triangle ABCABC is given together with an arbitrary circle ω\omega. Let α\alpha be the reflection of ω\omega with respect to AA, β\beta the reflection of ω\omega with respect to BB, and γ\gamma the reflection of ω\omega with respect to CC. It is known that the circles α,β,γ\alpha, \beta, \gamma don't intersect each other. Let PP be the meeting point of the two internal common tangents to β,γ\beta, \gamma (see picture). Similarly, QQ is the meeting point of the internal common tangents of α,γ\alpha, \gamma, and RR is the meeting point of the internal common tangents of α,β\alpha, \beta. Prove that the triangles PQR,ABCPQR, ABC are congruent.
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