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Let $\triangle ABC$ be an arbitrary triangle and let $G$ be its centroid. Three medians are denoted by $AD,BE,CF$. I am attempting to show that the circumcentres of $\triangle AGF,\triangle GFB,\triangle BGD,\triangle DGC,\triangle CGE,\triangle EGA$ lie on a circle. I've worked on the problem several days by some analytic means and find it hard to solve, so I wonder if there is an elegent way to prove the result, and is there a description of the center of this circle?

Any advise or help would be appreciated, thanks.

Phil. Z
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    Each triangle has 3 excenters, which one you are talking about? All three? – Saša Jul 14 '18 at 16:03
  • @Oldboy Sorry, that was a typo, it should be circumcentre. Thank you for pointing it out. – Phil. Z Jul 15 '18 at 00:39
  • Here is another Conjecture on Six Concyclic Circumcenter : https://commons.hostos.cuny.edu/mtrj/wp-content/uploads/sites/30/2022/04/v14n1-Problem-Corner.pdf –  Apr 16 '22 at 17:35

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The problem describes famous Van Lamoen circle.

The proof is not elementary (won't fit a single sheet of paper, I mean) and you can find several diferent ones on the web. Actually, this is the problem of the Olympic caliber. Two different proofs:

The Lamoen circle, Darij Grinberg

Another proof of van Lamoen’s Theorem and Its Converse, Nguyen Minh Ha

The hexagon $A_bA_cB_aB_cC_aC_b$ has other interesting properties. For example, the opposite sides are parallel and main diagonals are of equal length.

enter image description here

Saša
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