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 the Gilbert point of
T
, which is point
X
1157
in Clark Kimberling's list of triangle centers.
The theorem was proposed as an advanced problem by J. R. Musselman and R. Goormaghtigh in 1939, and a proof was presented by them in 1941. A generalization of this result was stated and proved by Goormaghtigh.
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
O
A
,
O
B
, and
O
C
, such that
O
A
′
/
O
A
=
O
B
′
/
O
B
=
O
C
′
/
O
C
=
t
. Consider the three lines
L
A
,
L
B
, and
L
C
, perpendicular to
O
A
,
O
B
, and
O
C
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
B
C
,
C
A
, and
A
B
, 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
. Let
N
be the projection of the circumcenter
O
on the line
R
, and
N
′
the point on
O
N
such that
O
N
′
/
O
N
=
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
O
H
, such that
Q
H
/
Q
O
=
2
t
.
