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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Geometricorum, volume 1.