# Musselman's theorem

Jump to navigation Jump to search

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 the Gilbert point of $T$ , which 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 $QH/QO=2t$ .