
Journal for Geometry and Graphics 27 (2023), No. 1, 011028 Copyright Heldermann Verlag 2023 Family of Conics Having Double Contact with two Intersecting Ellipses Anastasia Taouktsoglou Democritus University of Thrace, Xanthi, Greece ataoukts@pme.duth.gr George Lefkaditis Patras University, Patras, Greece glef@upatras.gr We prove using projective geometry, analytic geometry and calculus the converse of the theorem, which is proved with synthetic projective geometry in J. L. S. Hatton’s book The Principles of Projective Geometry Applied to the Straight Line and Conic, Cambridge University Press (1913), p. 287, case (b). This theorem, as well as its converse, refer to properties that exist when a conic C_{3} contacts two other intersecting conics C_{1} and C_{2} and specifically concern the existing harmonic pencil between common chords of C_{1}, C_{2} and the pair of their contact chords with C_{3}. With the proof of the converse theorem, which is achieved here in the case of two concentric ellipses, the problem of constructing a conic C_{3} is also addressed. In addition we investigate the type of conic C_{3}, which is tangent to C_{1}, C_{2}, and the condition that is required for C_{3} to be an ellipse, a hyperbola or a degenerate parabola, either inscribed or circumscribed to C_{1}, C_{2}. Finally, we refer to the existing involution between the common fixed chords and the changing contact chords. Keywords: Harmonic pencil, concentric ellipses, conjugate points, double contact conic, involution. MSC: 51N15; 51N20, 68U05. [ Fulltextpdf (2985 KB)] 