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.

See also

• Parry circle
• Lester circle