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]

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

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

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The van Lamoen circle is named after the mathematician Floor van Lamoen [nl] 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

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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

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References

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  1. ^ a b c Kimberling, Clark, Encyclopedia of Triangle Centers, retrieved 2014-10-10. See X(1153) = Center of the van Lemoen circle.
  2. ^ a b Weisstein, Eric W., "van Lamoen circle", MathWorld, retrieved 2014-10-10
  3. ^ van Lamoen, Floor (2000), Problem 10830, vol. 107, American Mathematical Monthly, p. 893
  4. ^ a b Li, Kin Y. (2001), "Concyclic problems" (PDF), Mathematical Excalibur, 6 (1): 1–2
  5. ^ (2002), Solution to Problem 10830. American Mathematical Monthly, volume 109, pages 396-397.
  6. ^ Myakishev, Alexey; Woo, Peter Y. (2003), "On the Circumcenters of Cevasix Configuration" (PDF), Forum Geometricorum, 3: 57–63
  7. ^ Ha, N. M. (2005), "Another Proof of van Lamoen's Theorem and Its Converse" (PDF), Forum Geometricorum, 5: 127–132