Consider three concentric ellipses Ei, i= 1, 2, 3, each defined by a pair of conjugate semi–diameters taken from a given bundle of three coplanar line segments (where only two of them are permitted to coincide). In a proof by GA Peschka of the Karl Pohlke’s Fundamental Theorem of Axonometry, a parallel projection of a sphere onto a plane, say E, is adopted to show that a new concentric (to Ei) ellipse E exists,“circumscribing” all Ei, ie E is simultaneously tangent to all Ei⊂ E, i= 1, 2, 3. Motivated by the above, this paper investigates the plane–geometric problem of determining all the existing circumscribing ellipses (like E) of Ei, i= 1, 2, 3, exclusively from the Analytic Plane Geometry’s point of view (unlike the sphere’s parallel projection that requires the adoption of a three–dimensional space). It is proved that, at most, two circumscribing ellipses (of Ei) exist. One of them is always existing while, under certain conditions, another circumscribing ellipse (of Ei), say E∗(= E), can also exist. Moreover, in case this second circumscribing ellipse E∗ does not exist, then a hyperbola (concentric to Ei) exists instead, and is (simultaneously) tangent to all Ei, i= 1, 2, 3. The above results and their calculations are demonstrated by various examples and figures.