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X lies on circumcircle of triangle formed by polars of X

Source: KoMaL A. 871

February 15, 2024
geometrycircumcirclepoles and polars

Problem Statement

Let ABCABC be an obtuse triangle, and let HH denote its orthocenter. Let ωA\omega_A denote the circle with center AA and radius AHAH. Let ωB\omega_B and ωC\omega_C be defined in a similar way. For all points XX in the plane of triangle ABCABC let circle Ω(X)\Omega(X) be defined in the following way (if possible): take the polars of point XX with respect to circles ωA\omega_A, ωB\omega_B and ωC\omega_C, and let Ω(X)\Omega(X) be the circumcircle of the triangle defined by these three lines. With a possible exception of finitely many points find the locus of points XX for which point XX lies on circle Ω(X)\Omega(X).
Proposed by Vilmos Molnár-Szabó, Budapest
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