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

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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