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]

History

The theorem was proposed as an advanced problem by John Rogers Musselman and René Goormaghtigh in 1939,^[4] and a proof was presented by them in 1941.^[5] A generalization of this result was stated and proved by Goormaghtigh.^[6]

Goormaghtigh’s generalization

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

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

It had been observed by Joseph Neuberg, in 1884, that the three points ${\displaystyle P_{A}}$, ${\displaystyle P_{B}}$, and ${\displaystyle P_{C}}$ lie on a common line ${\displaystyle R}$.^[7] Let ${\displaystyle N}$ be the projection of the circumcenter ${\displaystyle O}$ on the line ${\displaystyle R}$, and ${\displaystyle N'}$ the point on ${\displaystyle ON}$ such that ${\displaystyle ON'/ON=t}$. Goormaghtigh proved that ${\displaystyle N'}$ is the inverse with respect to the circumcircle of ${\displaystyle T}$ of the isogonal conjugate of the point ${\displaystyle Q}$ on the Euler line ${\displaystyle OH}$, such that ${\displaystyle QH/QO=2t}$.^[8][9]