Van Lamoen circle

In Euclidean 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

be its centroid (the intersection of its three medians).

be the midpoints of the sidelines

It turns out that the circumcenters of the six triangles

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.

[1][5] The center of the van Lamoen circle is point

in Clark Kimberling's comprehensive list of triangle 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

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

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.