
Journal for Geometry and Graphics 11 (2007), No. 1, 015026 Copyright Heldermann Verlag 2007 Another Cubic Associated with a Triangle Sadi AbuSaymeh Dept. of Mathematics, Yarmouk University, Irbid, Jordan sade@yu.edu.jo Mowaffaq Hajja Dept. of Mathematics, Yarmouk University, Irbid, Jordan mhajja@yu.edu.jo Hellmuth Stachel Inst. of Discrete Mathematics and Geometry, University of Technology, Wiedner Hauptstr. 810/104, Vienna, Austria stachel@dmg.tuwien.ac.at Let ABC be a triangle with sidelengths a, b, and c. For a point P in its plane, let AP_{a}, BP_{b}, and CP_{c} be the cevians through P. It was proved that the centroid, the Gergonne point, and the Nagel point are the only centers for which (the lengths of) BP_{a}, CP_{b}, and AP_{c} are linear forms in a, b, and c, i.e., for which [AP_{a} BP_{b} CP_{c}] = [a b c]L for some matrix L. In this note, we investigate the locus of those centers for which BP_{a}, CP_{b}, and AP_{c} are quasilinear in a, b, and c in the sense that they satisfy [AP_{a} BP_{b} CP_{c}]M = [a b c]L for some matrices L and M. We also see that the analogous problem of finding those centers for which the angles BAP_{a}, CBP_{b}, and ACP_{c} are quasilinear in the angles A, B, and C leads to what is known as the Balaton curve. Keywords: triangle geometry, cevians, Nagel point, Gergonne point, irreducible cubic, Balaton curve, perimeter trisecting points, sidebalanced triangle. MSC: 51M04; 51N35 [ Fulltextpdf (176 KB)] for subscribers only. 