Musselman's theorem

{{short description|About a common point of certain circles defined by an arbitrary triangle}}

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 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 $QH/QO = 2t$ .

References

Jean-Louis Ayme, [http://jl.ayme.pagesperso-orange.fr/Docs/Le%20point%20de%20Kosnitza.pdf le point de Kosnitza], page 10. Online document, accessed on 2014-10-05.

D. Grinberg (2003) [http://forumgeom.fau.edu/FG2003volume3/FG200311index.html On the Kosnita Point and the Reflection Triangle]. Forum Geometricorum, volume 3, pages 105–111

John Rogers Musselman and René Goormaghtigh (1939), Advanced Problem 3928. American Mathematical Monthly, volume 46, page 601

Clark Kimberling (2014), [http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X1157 Encyclopedia of Triangle Centers], section X(1157) . Accessed on 2014-10-08

John Rogers Musselman and René Goormaghtigh (1941), Solution to Advanced Problem 3928. American Mathematics Monthly, volume 48, pages 281–283

Eric W. Weisstein (), [http://mathworld.wolfram.com/MusselmansTheorem.html Musselman's theorem]. online document, accessed on 2014-10-05.

Khoa Lu Nguyen (2005), [http://forumgeom.fau.edu/FG2005volume5/FG200502.pdf A synthetic proof of Goormaghtigh's generalization of Musselman's theorem]. Forum Geometricorum, volume 5, pages 17–20

Ion Pătrașcu and Cătălin Barbu (2012), [http://ijgeometry.com/wp-content/uploads/2011/10/21.pdf Two new proofs of Goormaghtigh theorem]. International Journal of Geometry, volume 1, pages=10–19, {{ISSN|2247-9880}}

Joseph Neuberg (1884), Mémoir sur le Tetraèdre. According to Nguyen, Neuberg also states Goormaghtigh's theorem, but incorrectly.

Category:Theorems about triangles and circles