Journal for Geometry and Graphics 18 (2014), No. 2, 133--157
Copyright Heldermann Verlag 2014
Equicevian Points and Cubics of a Triangle
Sadi Abu-Saymeh
School of Natural Resources, German-Jordanian University, Amman, Jordan
sadi-abosaymeh@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. 8-10/104, 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 AAP, BBP, and CCP 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 so-called 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
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