X(2), X(7), X(8), X(80), X(320), X(369), X(519), X(903), X(908), X(3232)
X(3232) = isotomic conjugate of X(369)
Ga, Gb, Gc : vertices of the antimedial triangle
infinite points of the Mandart circum-ellipse
remark : X(369) and X(3232) are the 1st and 2nd trisected perimeter points
see also Table 42 for other curves passing through X(369)
In memoriam Cyril Parry
who left us on February, 13 2005
Let P = u : v : w be a point lying inside ABC and let A', B', C' be the vertices of its cevian triangle.
Let Sa = AB' + AC' = bw / (w+u) + cv / (v+u) and define Sb, Sc similarly. Sb = Sc if and only if P lies on a circumcubic Qa passing through Ga, Gb, Gc, the midpoint of BC and X(369), the 1st trisected perimeter point (see TCCT, p.267). The tangent at A passes through X(8). Two other cubics Qb, Qc are defined likewise. See figure 1.
Now, if Sa = BC' + CB', we obtain three similar circumcubics passing through Ga, Gb, Gc and X(3232), the 2nd trisected perimeter point. The cubic Qa is tangent at A to AG and meets BC at the cevian of X(7), the A-vertex of the intouch triangle. These three cubics are obviously the isotomic transforms of the previous cubics. See figure 2.
In both cases, these three cubics form a net containing K311 which therefore also passes through X(369) and X(3232). These two points are isotomic conjugates hence collinear with X(320), the pivot of K311. See figure 3.
K311 is the isotomic pivotal cubic pK(X2, X320). It meets the line at infinity at X(519) and two imaginary points which also lie on the Mandart circum-ellipse with center X(9), perspector X(1). The real asymptote is the line X(88)X(519).
K311 appears in a forthcoming paper by Sadi Abu-Saymeh, Mowaffaq Hajja, and Hellmuth Stachel, "Another cubic associated with the triangle" in Journal for Geometry and Graphics. See X(3218) in Clark's ETC.
Compare K311 and K455, a similar cubic.
The cevian (or anticevian) triangle of P and the Furhmann triangle are orthologic if and only if P lies on K311. One center of orthology lies on K510 and the other on a cubic passing through X(3), X(8), X(946) with very little interest.