InEuclidean plane geometry, the van Lamoen circle is a special circle associated with any given triangle . It contains the circumcenters of the six triangles that are defined inside
by its three medians.[1][2]
Specifically, let ,
,
be the verticesof
, 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
.[2]
The van Lamoen circle is named after the mathematician Floor van Lamoen [nl] who posed it as a problem in 2000.[3][4] A proof was provided by Kin Y. Li in 2001,[4] and the editors of the Amer. Math. Monthly in 2002.[1][5]
The center of the van Lamoen circle is point inClark Kimberling's comprehensive listoftriangle centers.[1]
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).[6] A simpler proof of this result was given by Nguyen Minh Ha in 2005.[7]