In a previous study of ours we considered two concentric conics $C_1,C_2$ intersecting at four points and we searched all conics having double contact with these two. As a solution we found an one-parameter family of conics, the so-called double contact conics of $C_1,C_2$. We noticed that this family creates a hyperbolic involution $f_{AB}$ on the pencil of lines through their common centre $O$, with double lines the lines of the common diameters $AC,BD$ of $C_1,C_2$. The lines of the contact diameters of every double contact conic $C_3$ with $C_1,C_2$ also correspond through $f_{AB}$.
In the present study we consider three concentric ellipses, mutually conjugate (i.e. starting with three coplanar line segments $OA,OB,OC$, we consider three concentric ellipses $C_1,C_2,C_3$ defined, so that every two of the line segments are conjugate semi-diameters of one ellipse) and we search all conics having double contact with these three ellipses. The problem of finding a fourth concentric ellipse circumscribed to all three is solved through the three-dimensional space by G.A.Peschka (1879) in his proof of K.Pohlke's Fundamental Theorem of Axonometry. Previous studies of ours, dealing with the problem as a two-dimensional one, proved with Analytic Plane Geometry that at most two solutions to the problem exist. The primary solution $T_1$ always exists and it is an ellipse. That's why the problem is referred as the Four Ellipses Problem. The secondary solution $T_2$, i.e. the second concentric conic circumscribed to all three, (if it exists) is an ellipse or a hyperbola. Furthermore, only $T_1$ was constructed using Synthetic Projective Plane Geometry in these studies.
The present study focuses on the investigation of the existence and on the construction of the secondary solution $T_2$ of the Four Ellipses Problem using methods of Synthetic Projective Plane Geometry, in particular the Theory of Involution.
First we proved with Analytic Plane Geometry that the common diameters of every couple of $C_1,C_2,C_3$ correspond through an involution $f$. In the sequel, criteria of Synthetic Projective Plane Geometry determine whether $f$ is hyperbolic or elliptic.
If $f$ is hyperbolic, then there exist exactly two conics $T_1,T_2$ concentric to $C_1,C_2,C_3$, that circumscribe $C_1,C_2,C_3$. The primary solution $T_1$, is always an ellipse, while the secondary solution $T_2$ is an ellipse, a hyperbola or a degenerate parabola, i.e. a pair of parallel lines. In any case, the common diameters of $T_1,T_2$ define the double lines of $f$.
If $f$ is elliptic, then there still exist two conics $T_1,T_2$ concentric to $C_1,C_2$, $C_3$, that have double contact with $C_1,C_2,C_3$. But this time only the primary solution $T_1$ is an ellipse circumscribed to $C_1,C_2,C_3$, while $T_2$ is an ellipse inscribed to $C_1,C_2,C_3$.
Regardless of whether $f$ is hyperbolic or elliptic, $T_2$ can now be constructed with Synthetic Projective Plane Geometry, using the already constructed $T_1$ and the involution $f$, since the contact diameters of $T_1,C_i$ and $T_2,C_i$, $i=1,2,3$ also correspond through $f$.