Forum Geometricorum, 2 (2002) 71--87.
Abstract: Let hexagon PQRSTU be inscribed in triangle A_1A_2A_3 (ordered counterclockwise) such that P and S are on line A_3A_1, Q and T are on line A_1A_2, and R and U are on line A_2A_3. If PQ, RS, and TU are respectively parallel to A_2A_3, A_1A_2, and A_3A_1, while QR, ST, and UP are antiparallel to A_3A_1, A_2A_3, and A_1A_2 respectively, the vertices of the hexagon are on one circle. Now, let hexagon P'Q'R'S'T'U' be described as above, with each of its sides parallel to the corresponding side of PQRSTU. Again the six vertices are concyclic, and the process may be repeated indefinitely to form an infinite family of circles (Tucker [3]). This family is a coaxaloid system, and its locus of centers is the Brocard axis of the triangle, passing through the circumcenter and the symmedian point. J. A. Third [2] extended this idea by relaxing the conditions for the directions of the sides of the hexagon, thus finding infinitely many coaxaloid systems of circles. The present paper defines a further extension by allowing the directions of the sides to be as arbitrary as possible, resulting in families of homothetic conics with properties analogous to those of the Tucker circles.
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