While working on a problem not exactly in plane geometry, I seem to have established the statement below. I wonder if any one has seen it before or can recognize it as equivalent to a known fact. It may be related to the Miquel point theorem, which came out in a related problem.
On the sides (and continuations) of triangle ABC take points Ac, Bc, ..., Ab :
on AB : Ac, Bc, s.t. A->Ac = B->Bc; on BC : Ba, Ca, s.t. B->Ba = C->Ca; on CA : Cb, Ab, s.t. C->Cb = A->Ab;
A->Ac is the vector (_directed_ segment) from A to Ac.
Let tA, tB, tC be the tangents to the circumcircles of AAbA, BBaBc, CCbCa at A, B, C.