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Journal for Geometry and Graphics 20 (2016), No. 1, 085--100
Copyright Heldermann Verlag 2016

Symbiotic Conics and Quartets of Four-Foci Orthogonal Circles

Maurizio Ternullo
INAF, Osservatorio Astrofisico, v. S. Sofia 78, 95123 Catania, Italia

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

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