# 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 the Gilbert point of ${\displaystyle T}$, which 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]

## References

1. ^ D. Grinberg (2003) On the Kosnita Point and the Reflection Triangle. Forum Geometricorum, volume 3, pages 105–111
2. ^ a b Eric W. Weisstein (), Musselman's theorem. online document, accessed on 2014-10-05.
3. ^ Clark Kimberling (2014), Encyclopedia of Triangle Centers, section X(1154) = Gilbert Point. Accessed on 2014-10-08
4. ^ John Rogers Musselman and René Goormaghtigh (1939), Advanced Problem 3928. American Mathematical Monthly, volume 46, page 601
5. ^ John Rogers Musselman and René Goormaghtigh (1941), Solution to Advanced Problem 3928. American Mathematics Monthly, volume 48, pages 281–283
6. ^ Jean-Louis Ayme, le point de Kosnitza, page 10. Online document, accessed on 2014-10-05.
7. ^ Joseph Neuberg (1884), Mémoir sur le Tetraèdre. According to Nguyen, Neuberg also states Goormaghtigh's theorem, but incorrectly.
8. ^ Khoa Lu Nguyen (2005), A synthetic proof of Goormaghtigh's generalization of Musselman's theorem. Forum Geometricorum, volume 5, pages 17–20
9. ^ Ion Pătrașcu and Cătălin Barbu (2012), Two new proofs of Goormaghtigh theorem. International Journal of Geometry, volume 1, pages=10–19, ISSN 2247-9880