I am looking a proof for the problem as follows:

Let a convex hexagon, such that its principal diagonals are concurrent. For each side of the hexagon, extend the adjacent sides to their intersection, forming a triangle exterior to the given side. Then show that: Three external (or internal) homothetic centers of three pair circumcircle of opposite triangles are collinear.

Please see the applet in Geogebra

PS: The line through the three external homothetic centers are perpendicular to the line through three internal homothetic centers.

Another applet: Three homothetic centers are collinear associated a circumscribed conic hexagon

convexhexagon ? and not valid for a general case. $\endgroup$ – ARi Jan 15 '17 at 18:13