IL-GILLIS 2023 Problem 3

A triangle $ABC$ is given together with an arbitrary circle $\omega$ . Let $\alpha$ be the reflection of $\omega$ with respect to $A$ , $\beta$ the reflection of $\omega$ with respect to $B$ , and $\gamma$ the reflection of $\omega$ with respect to $C$ . It is known that the circles $\alpha, \beta, \gamma$ don't intersect each other. Let $P$ be the meeting point of the two internal common tangents to $\beta, \gamma$ (see picture). Similarly, $Q$ is the meeting point of the internal common tangents of $\alpha, \gamma$ , and $R$ is the meeting point of the internal common tangents of $\alpha, \beta$ . Prove that the triangles $PQR, ABC$ are congruent.

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