
Journal for Geometry and Graphics 20 (2016), No. 1, 085100 Copyright Heldermann Verlag 2016 Symbiotic Conics and Quartets of FourFoci Orthogonal Circles Maurizio Ternullo INAF, Osservatorio Astrofisico, v. S. Sofia 78, 95123 Catania, Italia mternullo@oact.inaf.it A quartet of orthogonal circles  one of them being imaginary  associated with a general point P taken on a given ellipse H is described. The mutual intersections of these circles, their intersections with Barlotti's circles and further, newly introduced points are peculiar under several aspects. A major result is the finding of a complete, cyclic quadrangle having two diagonal points in fixed positions on the minor axis of the ellipse; these diagonal points are concyclic with the ellipse foci, in spite of the dependence of the whole figure from location of the point P. Two conics  the {symbiotic ellipse and hyperbola  are introduced, in association with P; such conics are characterized by the fact that they (i) have P as center and the tangent and normal to H at P as axes of symmetry, (ii) pass through the center H of the ellipse, and (iii) admit the axes of symmetry of the ellipse H as tangent and normal. Several relationships among these conics are described. The study of the symbiotic ellipse reveals new properties of the ellipse H. Keywords: Ellipse, Monge's circle, Barlotti's circles, concyclic points, collinear points, complete quadrangle. MSC: 51M04; 51N20 [ Fulltextpdf (312 KB)] 