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Consider three concentric ellipses E i, 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 G. A. 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 E i) ellipse E exists, “circumscribing” all E i, i.e. E is simultaneously tangent to all E i⊂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 E i, 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 E i) exist. One of them is always existing while, under certain conditions, another circumscribing ellipse (of E i), say E∗ ( 6=E), can also exist. Moreover, in case this second circumscribing ellipse E∗ does not exist, then a hyperbola (concentric to E i) exists instead, and is (simultaneously) tangent to all E i, i = 1,2,3. The above results and their calculations are demonstrated by various examples and figures.

Parallel Projected Sphere on a Plane: A New Plane–Geometric Investigation