# Isogonal image of the circumcircleWhat is it? A Mathematical Droodle

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Explanation

Let P be a point in the plane of ΔABC. Reflect the lines PA, PB and PC in the bisectors of angles A, B, and C, respectively. In most cases, the lines thus obtained are concurrent. The point Q of concurrency is known as the isogonal conjugate of P.

In fact, the three lines always intersect, not just in most cases, although some cases do appear exceptional. When P happens to lie on the circumcircle of ΔABC, its isogonal conjugate lies on the line at infinity. When P lies on the circumcircle, the reflections of PA, PB and PC become parallel, i. e., intersect at infinity.

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