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 by its three medians.
Specifically, let,, be the vertices of, and let be its centroid. 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, at which point the six circumcenters degenerate into the three Euler points of the nine-point circle. A simpler proof of this result was given by Nguyen Minh Ha in 2005.