Musselman's theorem explained
In Euclidean geometry, Musselman's theorem is a property of certain circles defined by an arbitrary triangle.
Specifically, let
be a triangle, and
,
, and
its
vertices. Let
,
, and
be the vertices of the reflection triangle
, obtained by mirroring each vertex of
across the opposite side. Let
be the circumcenter of
. Consider the three circles
,
, and
defined by the points
,
, and
, respectively. The theorem says that these three
Musselman circles meet in a point
, that is the inverse with respect to the circumcenter of
of the
isogonal conjugate or the
nine-point center of
.
The common point
is point
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
,
, and
be the vertices of a triangle
, and
its circumcenter. Let
be the orthocenter of
, that is, the intersection of its three
altitude lines. Let
,
, and
be three points on the segments
,
, and
, such that
. Consider the three lines
,
, and
, perpendicular to
,
, and
though the points
,
, and
, respectively. Let
,
, and
be the intersections of these perpendicular with the lines
,
, and
, respectively.
It had been observed by Joseph Neuberg, in 1884, that the three points
,
, and
lie on a common line
. Let
be the projection of the circumcenter
on the line
, and
the point on
such that
. Goormaghtigh proved that
is the inverse with respect to the circumcircle of
of the isogonal conjugate of the point
on the
Euler line
, such that
.
References
[1] [2] [3] [4] [5] [6] [7] [8] [9]
Notes and References
- Jean-Louis Ayme, le point de Kosnitza, page 10. Online document, accessed on 2014-10-05.
- D. Grinberg (2003) On the Kosnita Point and the Reflection Triangle. Forum Geometricorum, volume 3, pages 105–111
- [John Rogers Musselman]
- Clark Kimberling (2014), 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, Musselman's theorem. online document, accessed on 2014-10-05.
- Khoa Lu Nguyen (2005), A synthetic proof of Goormaghtigh's generalization of Musselman's theorem. Forum Geometricorum, volume 5, pages 17–20
- Ion Pătrașcu and Cătălin Barbu (2012), Two new proofs of Goormaghtigh theorem. International Journal of Geometry, volume 1, pages=10–19,
- [Joseph Jean Baptiste Neuberg|Joseph Neuberg]
