G1,G2,G3 = the centroids of A'B"C", B'C"A", C'A"B", resp.
g1,g2,g3 = the centroids of A"B'C', B"C'A', C"A'B', resp.
The circumcenter of the circle (G1G2G3) is the common circumcenter of A'B'C' and A"B"C", the N of ABC.
The circle (G1G2G3) passes through G.
1. The triangles G1G23, g1g2g3 are perspective.
The six centroids lie on a conic (rectangular hyperbola) with center P, the perspector of the triangles.
2. The NPC centers of the triangles G1g2g3, G2g3g1, G3g1g2, G1G2G3, g1g2g3 concur at P (center of the hyperbola)
Antreas P. Hatzipolakis, 12 May 2013