In Euclidean plane geometry, the van Lamoen circle is a special circle associated with any given triangle
T
. It contains the circumcenters of the six triangles that are defined inside
T
by its three medians.
Specifically, let
A
,
B
,
C
be the vertices of
T
, and let
G
be its centroid (the intersection of its three medians). Let
M
a
,
M
b
, and
M
c
be the midpoints of the sidelines
B
C
,
C
A
, and
A
B
, respectively. It turns out that the circumcenters of the six triangles
A
G
M
c
,
B
G
M
c
,
B
G
M
a
,
C
G
M
a
,
C
G
M
b
, and
A
G
M
b
lie on a common circle, which is the van Lamoen circle of
T
.
The van Lamoen circle is named after the mathematician Floor van Lamoen who posed it as a problem in 2000. A proof was provided by Kin Y. Li in 2001, and the editors of the Amer. Math. Monthly in 2002.
The center of the van Lamoen circle is point
X
(
1153
)
in Clark Kimberling's comprehensive list of triangle centers.
In 2003, Alexey Myakishev and Peter Y. Woo proved that the converse of the theorem is nearly true, in the following sense: let
P
be any point in the triangle's interior, and
A
A
′
,
B
B
′
, and
C
C
′
be its cevians, that is, the line segments that connect each vertex to
P
and are extended until each meets the opposite side. Then the circumcenters of the six triangles
A
P
B
′
,
A
P
C
′
,
B
P
C
′
,
B
P
A
′
,
C
P
A
′
, and
C
P
B
′
lie on the same circle if and only if
P
is the centroid of
T
or its orthocenter (the intersection of its three altitudes). A simpler proof of this result was given by Nguyen Minh Ha in 2005.
