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.^[1][2]

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 $T$.^[2]

History

The van Lamoen circle is named after the mathematician Floor van Lamoen who posed it as a problem in 2000.^[3][4] A proof was provided by Kin Y. Li in 2001,^[4] and the editors of the Amer. Math. Monthly in 2002.^[1][5]

Properties

The center of the van Lamoen circle is point $X(1153)$ in Clark Kimberling's comprehensive list of triangle centers.^[1]

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).^[6] A simpler proof of this result was given by Nguyen Minh Ha in 2005.^[7]

See also

• Parry circle
• Lester circle

References

1. Clark Kimberling (), X(1153) = Center of the van Lemoen circle, in the Encyclopedia of Triangle Centers Accessed on 2014-10-10.
2. Eric W. Weisstein, van Lamoen circle at Mathworld. Accessed on 2014-10-10.
3. Floor van Lamoen (2000), Problem 10830 American Mathematical Monthly, volume 107, page 893.
4. Kin Y. Li (2001), Concyclic problems. Mathematical Excalibur, volume 6, issue 1, pages 1-2.
5. (2002), Solution to Problem 10830. American Mathematical Monthly, volume 109, pages 396-397.
6. Alexey Myakishev and Peter Y. Woo (2003), [http://forumgeom.fau.edu/F*2003volume3/FG200305.pdf On the Circumcenters of Cevasix Configuration. Forum Geometricorum, volume 3, pages 57-63.
7. N. M. Ha (2005), Another Proof of van Lamoen's Theorem and Its Converse. Forum Geometricorum, volume 5, pages 127-132.