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Similar triangles and equal segments

Source: BMO SL 2023 G2

May 3, 2024
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Problem Statement

Let ABCDABCD be a cyclic quadrilateral with circumcenter OO lying in the interior. Let EE and FF be the midpoints of the segments BCBC and ADAD, respectively. Let XX be the point lying on the same side of the line EFEF as the vertex CC such that △EXF\triangle EXF and △BOA\triangle BOA are similar. Prove that XC=XDXC = XD.
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