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van Lamoen Circle

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vanLamoenCircle

The van Lamoen circle is the circle through the circumcenters of the six smaller triangles formed by the three medians of a reference triangle. The six circumcircles are shown in blue above. Their circumcenters O_(AB), O_(BA), etc. are concyclic, with the van Lamoen circle shown in green.

It has circle function

l=-((a^2-2b^2-2c^2)(8a^4-20a^2b^2+8b^4-20a^2c^2-11b^2c^2+8c^4)R^2)/(108a^2b^3c^3),
(1)

where R is the circumradius of the reference triangle.

Its center is Kimberling center X_(1153), which has triangle center function

alpha_(1153)=(10a^4-13a^2b^2+4b^4-13a^2c^2-10b^2c^2+4c^4)/a.
(2)

Its radius is

R_V=(sqrt((a^2-2b^2-2c^2)(2a^2+2b^2-c^2)(2a^2-b^2+2c^2)(2a^4-5a^2b^2+2b^4-5a^2c^2-5b^2c^2+2c^4))R^2)/(18a^2b^2c^2).
(3)

No Kimberling centers lie on the van Lamoen circle.

See also

Central Circle, Triangle Median

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References

Li, K. Y. "Concyclic Problems." Math. Excalibur, 6-1, 1-2, 2001. https://www.math.hkust.edu.hk/excalibur/v6_n1.pdf.Myakishev, A. and Woo, P. "On the Circumcenters of Cevasix Configurations." Forum Geom. 3, 57-63, 2003. https://web.archive.org/web/20240531131006/https://forumgeom.fau.edu/FG2003volume3/FG200305index.html.van Lamoen, F. "Problem 10830." Amer. Math. Monthly 107, 863, 2000.van Lamoen, F. "Solution to Problem 10830." Amer. Math. Monthly 109, 396-397, 2002.

Referenced on Wolfram|Alpha

van Lamoen Circle

Cite this as:

Weisstein, Eric W. "van Lamoen Circle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/vanLamoenCircle.html

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