Forum Geometricorum, 7 (2007) 169--180.

Abstract: The intersections of the corresponding sidelines of the cevian triangles of two points P_0 and P_1 form the anticevian triangle of a point T(P_0, P_1). We prove a number of interesting results relating the pair of inscribed conics with perspectors (Brianchon points) P_0 and P_1, in particular, a simple description of the fourth common tangent of the conics. We also show that the corresponding sides of the cevian triangles of points are concurrent if and only if the points lie on a circumconic. A characterization is given of circumconics whose centers lie on the cevian circumcircles of points on them (Brianchon - Poncelet theorem). We also construct a number of new triangle centers with very simple coordinates.

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