Musselman's theorem

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



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

The common point $$M$$ is point $$X_{1157}$$ in Clark Kimberling's list of triangle centers.

History
The theorem was proposed as an advanced problem by John Rogers Musselman and René Goormaghtigh in 1939, and a proof was presented by them in 1941. A generalization of this result was stated and proved by Goormaghtigh.

Goormaghtigh’s generalization
The generalization of Musselman's theorem by Goormaghtigh does not mention the circles explicitly.

As before, let $$A$$, $$B$$, and $$C$$ be the vertices of a triangle $$T$$, and $$O$$ its circumcenter. Let $$H$$ be the orthocenter of $$T$$, that is, the intersection of its three altitude lines. Let $$A'$$, $$B'$$, and $$C'$$ be three points on the segments $$OA$$, $$OB$$, and $$OC$$, such that $$OA'/OA=OB'/OB=OC'/OC = t$$. Consider the three lines $$L_A$$, $$L_B$$, and $$L_C$$, perpendicular to $$OA$$, $$OB$$, and $$OC$$ though the points $$A'$$, $$B'$$, and $$C'$$, respectively. Let $$P_A$$, $$P_B$$, and $$P_C$$ be the intersections of these perpendicular with the lines $$BC$$, $$CA$$, and $$AB$$, respectively.

It had been observed by Joseph Neuberg, in 1884, that the three points $$P_A$$, $$P_B$$, and $$P_C$$ lie on a common line $$R$$. Let $$N$$ be the projection of the circumcenter $$O$$ on the line $$R$$, and $$N'$$ the point on $$ON$$ such that $$ON'/ON = t$$. Goormaghtigh proved that $$N'$$ is the inverse with respect to the circumcircle of $$T$$ of the isogonal conjugate of the point $$Q$$ on the Euler line $$OH$$, such that $$QH/QO = 2t$$.