Geodesic
Another Cubic Associated with a Triangle
Journal for geometry and graphics, Tome 11 (2007) no. 1, pp. 15-26 Cet article a éte moissonné depuis la source Heldermann Verlag

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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.
Classification : 51M04, 51N35
Mots-clés : triangle geometry, cevians, Nagel point, Gergonne point, irreducible cubic, Balaton curve, perimeter trisecting points, side-balanced triangle 
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     title = {Another {Cubic} {Associated} with a {Triangle}},
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S. Abu-Saymeh; M. Hajja; H. Stachel. Another Cubic Associated with a Triangle. Journal for geometry and graphics, Tome 11 (2007) no. 1, pp. 15-26. http://geodesic.mathdoc.fr/item/JGG_2007_11_1_JGG_2007_11_1_a1/