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Many Feuerbach points and similar quads

Source: Israel 2020 Olympic Revenge

March 22, 2021
geometryFeuerbach

Problem Statement

Original post by shalomrav, but for some reason the mods locked the problem without any solves :noo:
Let ABCDABCD be a cyclic quadrilateral inscribed in circle Ω\Omega. Let FAF_A be the (associated with Ω\Omega) Feuerbach point of the triangle formed by the tangents to Ω\Omega at B,C,DB,C,D, that is, the point of tangency of Ω\Omega and the nine-point circle of that triangle. Define FB,FC,FDF_B, F_C, F_D similarly. Let A′A' be the intersection of the tangents to Ω\Omega at AA and FAF_A. Define B′,C′,D′B', C', D' similarly. Prove that quadrilaterals ABCDABCD and A′B′C′D′A'B'C'D' are similar
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