In Euclidean plane geometry, the van Lamoen circle is a special circle associated with any given triangle . It contains the circumcenters of the six triangles that are defined inside by its three medians. [1] [2]
Specifically, let , , be the vertices of , and let be its centroid (the intersection of its three medians). Let , , and be the midpoints of the sidelines , , and , respectively. It turns out that the circumcenters of the six triangles , , , , , and lie on a common circle, which is the van Lamoen circle of . [2]
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]
The center of the van Lamoen circle is point 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 be any point in the triangle's interior, and , , and be its cevians, that is, the line segments that connect each vertex to and are extended until each meets the opposite side. Then the circumcenters of the six triangles , , , , , and lie on the same circle if and only if is the centroid of 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]
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