Van Lamoen circle The van Lamoen circle through six circumcenters $A_{b}$ , $A_{c}$ , $B_{c}$ , $B_{a}$ , $C_{a}$ , $C_{b}$ 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 $BC$ , $CA$ , and $AB$ , respectively. It turns out that the circumcenters of the six triangles $AGM_{c}$ , $BGM_{c}$ , $BGM_{a}$ , $CGM_{a}$ , $CGM_{b}$ , and $AGM_{b}$ lie on a common circle, which is the van Lamoen circle of $T$ .

History

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.

Properties

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 $AA'$ , $BB'$ , and $CC'$ 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 $APB'$ , $APC'$ , $BPC'$ , $BPA'$ , $CPA'$ , and $CPB'$ 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.