Van Lamoen circle

The van Lamoen circle through six circumcenters , , , , ,

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]

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 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]

See also

References

  1. 1 2 3 Clark Kimberling (), X(1153) = Center of the van Lemoen circle, in the Encyclopedia of Triangle Centers Accessed on 2014-10-10.
  2. 1 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. 1 2 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.
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