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.
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.