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The Miquel Point "Returns"

Source: 2013 USAJMO #3/USAMO #1

April 30, 2013
geometrycircumcircleratioUSA(J)MOUSAMOSpiral Similaritygeometry solved

Problem Statement

In triangle ABCABC, points PP, QQ, RR lie on sides BCBC, CACA, ABAB respectively. Let ωA\omega_A, ωB\omega_B, ωC\omega_C denote the circumcircles of triangles AQRAQR, BRPBRP, CPQCPQ, respectively. Given the fact that segment APAP intersects ωA\omega_A, ωB\omega_B, ωC\omega_C again at XX, YY, ZZ, respectively, prove that YX/XZ=BP/PCYX/XZ=BP/PC.
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