Divide a triangle by its three medians into six smaller triangles. Surprisingly, the circumcenters circumcircles
of these smaller triangles (shown in blue above) are concyclic .
Their circumcircle (shown in green above) is known as the van Lamoen circle.
It has circle function
(1)
where circumradius of the reference
triangle .
Its center is Kimberling center
(2)
Its radius is
(3)
No Kimberling centers lie on the van Lamoen circle.
See also Central Circle ,
Triangle
Median
Explore with Wolfram|Alpha
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 https://mathworld.wolfram.com/vanLamoenCircle.html
Subject classifications
, 
, etc. of the six circumcircles
of these smaller triangles (shown in blue above) are concyclic.
Their circumcircle (shown in green above) is known as the van Lamoen circle.
is the circumradius of the reference
triangle.
, which has triangle center function
