
Journal for Geometry and Graphics 28 (2024), No. 1, 055072 Copyright by the authors licensed under CC BY SA 4.0 Double Contact Conics in Involution George Lefkaditis Patras University, Patras, Greece, Patras, Greece glef@upatras.gr Anastasia Taouktsoglou Democritus University of Thrace, Xanthi, Greece, Xanthi, Greece ataoukts@pme.duth.gr [Abstractpdf] Three coplanar line segments $OA$, $OB$, $OC$ are given and three concentric ellipses $C_1$, $C_2$, $C_3$ are defined, so that every two of the segments are conjugate semidiameters of one ellipse. In previous studies we proved using Analytic Plane Geometry that the problem of finding an ellipse circumscribed to $C_1$, $C_2$, $C_3$ has at most two solutions. The \emph{primary solution $T_1$} is always an ellipse. The \emph{secondary solution $T_2$} (if it exists) is an ellipse or a hyperbola. We also constructed $T_1$ using Synthetic Projective Plane Geometry.\par This study investigates the existence and the construction of $T_2$ with Synthetic Projective Geometry, particularly Theory of Involution. We prove that the common diameters of every couple of $C_1$, $C_2$, $C_3$ correspond through an involution $f$. Criteria of Synthetic Projective Geometry determine whether $f$ is hyperbolic or elliptic. If $f$ is hyperbolic, exactly two double contact conics $T_1$, $T_2$ exist circumscribed to $C_1$, $C_2$, $C_3$. $T_1$ is always an ellipse. $T_2$ is an ellipse, a hyperbola or a degenerate parabola. The common diameters of $T_1$, $T_2$ define the double lines of $f$. If $f$ is elliptic, still two double contact conics $T_1$, $T_2$ exist. Now $T_1$ is an ellipse circumscribed and $T_2$ an ellipse inscribed to $C_1$, $C_2$, $C_3$. Regardless of whether $f$ is hyperbolic or elliptic, we construct $T_2$ using the already constructed ellipse $T_1$ and the involution $f$. Keywords: Mutually conjugate ellipses, double contact conic, elliptic/hyperbolic involution, double rays, Fregier point. MSC: 51N15; 51N20, 68U05. [ Fulltextpdf (9151 KB)] 