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Topic: triangle center
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Posts: 5
Registered: 7/24/07
triangle center
Posted: Jul 24, 2007 7:16 PM
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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.

Then tA, tB, tC are concurrent or all parallel.

Dimiter Zlatanov

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