Clark Kimberling, Cubics Associated with Triangles of Equal Areas,

Forum Geometricorum, 1 (2001) 161 -- 170.

Abstract:  The locus of a point X for which the cevian triangle of X and that of its isogonal conjugate have equal areas is a cubic that passes through the 1st and 2nd Brocard points.  Generalizing from isogonal conjugate to P-isoconjugate yields a cubic Z(U,P) passing through U; if X is on Z(U,P), then the P-isoconjugate of X is on Z(U,P) and this point is collinear with X and U.  A generalized equal areas cubic \Gamma (P) is presented as a special case of Z(U,P). If \sigma = area(\triangle ABC), then the locus of X whose cevian triangle has prescribed oriented area K\sigma  is a cubic \Lambda (P), and P is  determined if K has a certain form. Various points are proved to lie on \Lambda (P).

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