Van Lamoen circle explained
. It contains the circumcenters of the six triangles that are defined inside
by its three
medians.
Specifically, let
,
,
be the
vertices of
, and let
be its
centroid (the intersection of its three medians). Let
,
, and
be the midpoints of the sidelines
,
, and
, respectively. It turns out that the circumcenters of the six triangles
,
,
,
,
, and
lie on a common circle, which is the van Lamoen circle of
.
History
The van Lamoen circle is named after the mathematician 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.
Properties
The center of the van Lamoen circle is point
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
be any point in the triangle's interior, and
,
, and
be its
cevians, that is, the
line segments that connect each vertex to
and are extended until each meets the opposite side. Then the circumcenters of the six triangles
,
,
,
,
, and
lie on the same circle if and only if
is the centroid of
or its orthocenter (the intersection of its three
altitudes). A simpler proof of this result was given by Nguyen Minh Ha in 2005.
See also
