Van Lamoen circle

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 https://nl.wikipedia.org/wiki/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

References

Eric W. Weisstein,
van Lamoen circle
' at Mathworld. Accessed on 2014-10-10.

Floor van Lamoen (2000), ''Problem 10830'' American Mathematical Monthly, volume 107, page 893.

(2002), ''Solution to Problem 10830''. American Mathematical Monthly, volume 109, pages 396-397.

Kin Y. Li (2001),
Concyclic problems
'. Mathematical Excalibur, volume 6, issue 1, pages 1-2.

Alexey Myakishev and Peter Y. Woo (2003),
On the Circumcenters of Cevasix Configuration
'. Forum Geometricorum, volume 3, pages 57-63.

N. M. Ha (2005),
Another Proof of van Lamoen's Theorem and Its Converse
'. Forum Geometricorum, volume 5, pages 127-132.

Clark Kimberling (),

', in the ''Encyclopedia of Triangle Centers'' Accessed on 2014-10-10.

Circles defined for a triangle