George E. Lefkaditis, Thomas L. Toulias, and Stelios Markatis,
On the circumscribing ellipse of three concentric ellipses,
Forum Geometricorum, 17 (2017) 527--547.
Abstract. Consider three coplanar non-degenerate line segments OA, OB, and OC, where only two of them can be collinear. Three concentric ellipses are then formed, say c'_1, c'_2, and c'_3, where (OA,OB), (OB,OC) and (OC,OA) are being respectively the corresponding three pairs of their defining conjugate semi-diameters. Then, there exist another concentric ellipse c* which circumscribes (i.e.\ being tangent to) all the ellipses c'_i, i = 1,2,3. Moreover, the common tangent line on each common (contact) point between each c'_i and their tangent ellipse c*, is parallel to the line segment (from the bundle of OA, OB, and OC) which does not belong to the pair of conjugate semi-diameters which forms each time the specific c'_i. The above result is derived through synthetic methods of the Projective Plane Geometry. Moreover, certain geometric properties (concerning, among others, the orthoptic circle of c* or the existence of an involution between two bundles of rays of c*), as well as the study of some special cases, are also discussed. A series of figures clarify the performed geometric constructions.
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