
Journal for Geometry and Graphics 18 (2014), No. 2, 133157 Copyright Heldermann Verlag 2014 Equicevian Points and Cubics of a Triangle Sadi AbuSaymeh School of Natural Resources, GermanJordanian University, Amman, Jordan sadiabosaymeh@gju.edu.jo Mowaffaq Hajja Mathematics Department, Yarmouk University, Irbid, Jordan mowhajja@yahoo.com Hellmuth Stachel Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstr. 810/104, 1230 Vienna, Austria stachel@dmg.tuwien.ac.at A point P in the plane of a given triangle ABC is said to be equicevian if the cevians AA_{P}, BB_{P}, and CC_{P} through P are of equal length. In this note, we see that the set Ω of equicevian points can be obtained via three cubic curves, and we give a complete description of Ω including also the imaginary solutions. There exist up to ten equicevian points, among them the four focal points of the Steiner circumellipse. Besides, we present properties of the socalled equicevian cubics which in the irreducible case are strophoids, i.e., rational and circular with orthogonal tangents at their node. Keywords: Equicevian points, equicevian cubics, strophoid, Steiner's circumellipse, focal points, focal curves, pedal curves, Marden's Theorem, Euclidean construction. MSC: 51N20; 51M25, 51M15 [ Fulltextpdf (354 KB)] for subscribers only. 