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 $AG{M}_c$, $BG{M}_c$, $BG{M}_a$, $CG{M}_a$, $CG{M}_b$, and $AG{M}_b$ lie on a common circle, which is the van Lamoen circle of $T$.

History

The van Lamoen circle is named after the mathematician Template:Ill 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 $A{A}^{\prime }$, $B{B}^{\prime }$, and $C{C}^{\prime }$ 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 $AP{B}^{\prime }$, $AP{C}^{\prime }$, $BP{C}^{\prime }$, $BP{A}^{\prime }$, $CP{A}^{\prime }$, and $CP{B}^{\prime }$ 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