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]}

References

1. D. Grinberg (2003) On the Kosnita Point and the Reflection Triangle . Forum Geometricorum, volume 3, pages 105–111
2. Eric W. Weisstein (), Musselman's theorem . online document, accessed on 2014-10-05.
3. Clark Kimberling (2014), Encyclopedia of Triangle Centers , section X(1157) . 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