Musselman's theorem

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.^[1] 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 $T$.^[2]

The common point $M$ is the Gilbert point of $T$, which is point $X_{1157}$ in Clark Kimberling's list of triangle centers.^[2][3]

History

The theorem was proposed as an advanced problem by J. R. Musselman and R. 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 $A$, $B$, 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 J. Neuberg, in 1884, that the three points $P_A$, $P_B$, and $P_C$ lie on a common line $R$.^[7] 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 $QH/QO = 2t$.^[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(1154) = Gilbert Point. Accessed on 2014-10-08
4. J. R. Musselman and R. Goormaghtigh (1939), Advanced Problem 3928. American Mathematics Monthly, volume 46, page 601
5. J. R. Musselman and R. 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. J. 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 Patrascu and Catalin Barbu (2012), Two new proofs of Goormaghtigh theorem. International Journal of Geometry, volume 1, pages=10–19, ISSN 2247-9880