Musselman's theorem

In Euclidean geometry, Musselman's theorem is a property of certain circles defined by an arbitrary triangle.

Specifically, let ${\displaystyle T}$ be a triangle, and ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ its vertices. Let ${\displaystyle A^{*}}$, ${\displaystyle B^{*}}$, and ${\displaystyle C^{*}}$ be the vertices of the reflection triangle ${\displaystyle T^{*}}$, obtained by mirroring each vertex of ${\displaystyle T}$ across the opposite side.^[1] Let ${\displaystyle O}$ be the circumcenter of ${\displaystyle T}$. Consider the three circles ${\displaystyle S_{A}}$, ${\displaystyle S_{B}}$, and ${\displaystyle S_{C}}$ defined by the points ${\displaystyle A\,O\,A^{*}}$, ${\displaystyle B\,O\,B^{*}}$, and ${\displaystyle C\,O\,C^{*}}$, respectively. The theorem says that these three Musselman circles meet in a point ${\displaystyle M}$, that is the inverse with respect to the circumcenter of ${\displaystyle T}$ of the isogonal conjugate or the nine-point center of ${\displaystyle T}$.^[2]

The common point ${\displaystyle M}$ is point ${\displaystyle X_{1157}}$ in Clark Kimberling's list of triangle centers.^[2][3]