Musselman's theorem


In Euclidean geometry, Musselman's theorem is a property of certain circles defined by an arbitrary triangle.
Specifically, let be a triangle, and,, and its vertices. Let,, and be the vertices of the reflection triangle, obtained by mirroring each vertex of across the opposite side. Let be the circumcenter of. Consider the three circles,, and defined by the points,, and, respectively. The theorem says that these three Musselman circles meet in a point, that is the inverse with respect to the circumcenter of of the isogonal conjugate or the nine-point center of.
The common point is point 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,, and be the vertices of a triangle, and its circumcenter. Let be the orthocenter of, that is, the intersection of its three altitude lines. Let,, and be three points on the segments,, and, such that. Consider the three lines,, and, perpendicular to,, and though the points ,, and, respectively. Let,, and be the intersections of these perpendicular with the lines,, and, respectively.
It had been observed by Joseph Neuberg, in 1884, that the three points,, and lie on a common line. Let be the projection of the circumcenter on the line, and the point on such that. Goormaghtigh proved that is the inverse with respect to the circumcircle of of the isogonal conjugate of the point on the Euler line, such that.