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Journal for Geometry and Graphics 11 (2007), No. 1, 015--026
Copyright Heldermann Verlag 2007

Another Cubic Associated with a Triangle

Sadi Abu-Saymeh
Dept. of Mathematics, Yarmouk University, Irbid, Jordan

Mowaffaq Hajja
Dept. of Mathematics, Yarmouk University, Irbid, Jordan

Hellmuth Stachel
Inst. of Discrete Mathematics and Geometry, University of Technology, Wiedner Hauptstr. 8-10/104, Vienna, Austria

Let ABC be a triangle with side-lengths a, b, and c. For a point P in its plane, let APa, BPb, and CPc 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) BPa, CPb, and APc are linear forms in a, b, and c, i.e., for which [APa BPb CPc] = [a b c]L for some matrix L. In this note, we investigate the locus of those centers for which BPa, CPb, and APc are quasi-linear in a, b, and c in the sense that they satisfy [APa BPb CPc]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 BAPa, CBPb, and ACPc are quasi-linear 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, side-balanced triangle.

MSC: 51M04; 51N35

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