X(2), X(7), X(8), X(80), X(320), X(369), X(519), X(903), X(908), X(3232), X(6224), X(8046), X(30578), X(34234), X(36917), X(36918) 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) other points below
 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). The isogonal transform of K311 is K312 = pK(X32, X36). The complement of K311 is K453 = pK(X44, X2). 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. See the related central cubic K510 in the page central cubics and also Q045, the trisected perimeter quartic. *** Locus properties 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. K311 is the locus of pivots of pivotal cubics pK(Ω, P) passing through X(369), Z1, Z2 and also X(519). See Table 42. The locus of poles is K1149 and the locus of isopivots is K1150. *** Other points on K311 T0=(a^4-2 a^2 b^2+b^4-2 a^3 c+a^2 b c+a b^2 c-2 b^3 c+2 a^2 c^2-3 a b c^2+2 b^2 c^2+2 a c^3+2 b c^3-3 c^4) (a^4-2 a^3 b+2 a^2 b^2+2 a b^3-3 b^4+a^2 b c-3 a b^2 c+2 b^3 c-2 a^2 c^2+a b c^2+2 b^2 c^2-2 b c^3+c^4): : ,SEARCH=8.281898902202453 T1=(a^2-b^2+4 b c-c^2) (a^2+2 a b+b^2-7 a c+2 b c+c^2) (a^2-7 a b+b^2+2 a c+2 b c+c^2): : ,SEARCH=-3.313611414225228 T2=(a^2-4 a b+b^2-c^2) (a^2-b^2-4 a c+c^2) (a^2+2 a b+b^2+2 a c-7 b c+c^2): : ,SEARCH=-17.08178572006066 T3=(a+b-c) (a-b+c) (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c+3 a^4 b c-a^3 b^2 c-3 a^2 b^3 c+3 a b^4 c-a^4 c^2-a^3 b c^2+4 a^2 b^2 c^2-a b^3 c^2-b^4 c^2+4 a^3 c^3-3 a^2 b c^3-a b^2 c^3-a^2 c^4+3 a b c^4-b^2 c^4-2 a c^5+c^6): : ,SEARCH=-0.5338129970938142 T4=(a+b-2 c) (a-2 b+c) (3 a^6-4 a^5 b-3 a^4 b^2+10 a^3 b^3-a^2 b^4-6 a b^5+b^6-4 a^5 c+8 a^4 b c-4 a^3 b^2 c-10 a^2 b^3 c+8 a b^4 c+2 b^5 c-3 a^4 c^2-4 a^3 b c^2+9 a^2 b^2 c^2-b^4 c^2+10 a^3 c^3-10 a^2 b c^3-4 b^3 c^3-a^2 c^4+8 a b c^4-b^2 c^4-6 a c^5+2 b c^5+c^6): : ,SEARCH=-0.8641338548553688 T5=(a+b-5 c) (a-5 b+c) (3 a^3-a^2 b-3 a b^2+b^3-a^2 c+2 a b c-b^2 c-3 a c^2-b c^2+c^3): : ,SEARCH=6.493766222529359 T6=-(a+b-5 c) (a-5 b+c) (7 a^3-3 a^2 b-9 a b^2+b^3-3 a^2 c+9 a b c+3 b^2 c-9 a c^2+3 b c^2+c^3): : ,SEARCH=8.919004738702797 T7=(a^2-a b+b^2-c^2) (a^2-b^2-a c+c^2) (a^8-4 a^6 b^2+6 a^4 b^4-4 a^2 b^6+b^8+6 a^5 b^2 c-2 a^4 b^3 c-10 a^3 b^4 c+6 a^2 b^5 c+4 a b^6 c-4 b^7 c-4 a^6 c^2+6 a^5 b c^2-11 a^4 b^2 c^2+8 a^3 b^3 c^2+9 a^2 b^4 c^2-14 a b^5 c^2+6 b^6 c^2-2 a^4 b c^3+8 a^3 b^2 c^3-20 a^2 b^3 c^3+10 a b^4 c^3+4 b^5 c^3+6 a^4 c^4-10 a^3 b c^4+9 a^2 b^2 c^4+10 a b^3 c^4-14 b^4 c^4+6 a^2 b c^5-14 a b^2 c^5+4 b^3 c^5-4 a^2 c^6+4 a b c^6+6 b^2 c^6-4 b c^7+c^8): : ,SEARCH=-6.216601117091038 T8=(a^2+2 a b+b^2-7 a c+2 b c+c^2) (a^2-7 a b+b^2+2 a c+2 b c+c^2) (a^6+4 a^5 b+5 a^4 b^2-5 a^2 b^4-4 a b^5-b^6+4 a^5 c-44 a^4 b c+18 a^3 b^2 c+34 a^2 b^3 c-20 a b^4 c+12 b^5 c+5 a^4 c^2+18 a^3 b c^2-57 a^2 b^2 c^2+14 a b^3 c^2+3 b^4 c^2+34 a^2 b c^3+14 a b^2 c^3-20 b^3 c^3-5 a^2 c^4-20 a b c^4+3 b^2 c^4-4 a c^5+12 b c^5-c^6): : ,SEARCH=2.369748856374253 T9=(a^2-4 a b+b^2-c^2) (a^2-b^2-4 a c+c^2) (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c+9 a^4 b c-4 a^3 b^2 c-6 a^2 b^3 c+6 a b^4 c-3 b^5 c-a^4 c^2-4 a^3 b c^2+10 a^2 b^2 c^2-4 a b^3 c^2-b^4 c^2+4 a^3 c^3-6 a^2 b c^3-4 a b^2 c^3+6 b^3 c^3-a^2 c^4+6 a b c^4-b^2 c^4-2 a c^5-3 b c^5+c^6): : ,SEARCH=5.95543317703446 tT3=(a-b-c) (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6+3 a^4 b c-3 a^3 b^2 c-a^2 b^3 c+3 a b^4 c-2 b^5 c-a^4 c^2-a^3 b c^2+4 a^2 b^2 c^2-a b^3 c^2-b^4 c^2-a^2 b c^3-3 a b^2 c^3+4 b^3 c^3-a^2 c^4+3 a b c^4-b^2 c^4-2 b c^5+c^6) (a^6-a^4 b^2-a^2 b^4+b^6-2 a^5 c+3 a^4 b c-a^3 b^2 c-a^2 b^3 c+3 a b^4 c-2 b^5 c-a^4 c^2-3 a^3 b c^2+4 a^2 b^2 c^2-3 a b^3 c^2-b^4 c^2+4 a^3 c^3-a^2 b c^3-a b^2 c^3+4 b^3 c^3-a^2 c^4+3 a b c^4-b^2 c^4-2 a c^5-2 b c^5+c^6): : ,SEARCH=6.230534184926302 tT4=(2 a-b-c) (a^6-6 a^5 b-a^4 b^2+10 a^3 b^3-3 a^2 b^4-4 a b^5+3 b^6+2 a^5 c+8 a^4 b c-10 a^3 b^2 c-4 a^2 b^3 c+8 a b^4 c-4 b^5 c-a^4 c^2+9 a^2 b^2 c^2-4 a b^3 c^2-3 b^4 c^2-4 a^3 c^3-10 a b^2 c^3+10 b^3 c^3-a^2 c^4+8 a b c^4-b^2 c^4+2 a c^5-6 b c^5+c^6) (a^6+2 a^5 b-a^4 b^2-4 a^3 b^3-a^2 b^4+2 a b^5+b^6-6 a^5 c+8 a^4 b c+8 a b^4 c-6 b^5 c-a^4 c^2-10 a^3 b c^2+9 a^2 b^2 c^2-10 a b^3 c^2-b^4 c^2+10 a^3 c^3-4 a^2 b c^3-4 a b^2 c^3+10 b^3 c^3-3 a^2 c^4+8 a b c^4-3 b^2 c^4-4 a c^5-4 b c^5+3 c^6): : ,SEARCH=-41.24669559351829 tT5=(5 a-b-c) (a^3-3 a^2 b-a b^2+3 b^3-a^2 c+2 a b c-b^2 c-a c^2-3 b c^2+c^3) (a^3-a^2 b-a b^2+b^3-3 a^2 c+2 a b c-3 b^2 c-a c^2-b c^2+3 c^3): : ,SEARCH=-0.6156613889770281 tT6=(5 a-b-c) (a^3-9 a^2 b-3 a b^2+7 b^3+3 a^2 c+9 a b c-3 b^2 c+3 a c^2-9 b c^2+c^3) (a^3+3 a^2 b+3 a b^2+b^3-9 a^2 c+9 a b c-9 b^2 c-3 a c^2-3 b c^2+7 c^3): : ,SEARCH=3.24597263050226 tT7=(a^2-b^2+b c-c^2) (a^8-4 a^7 b+6 a^6 b^2+4 a^5 b^3-14 a^4 b^4+4 a^3 b^5+6 a^2 b^6-4 a b^7+b^8+4 a^6 b c-14 a^5 b^2 c+10 a^4 b^3 c+10 a^3 b^4 c-14 a^2 b^5 c+4 a b^6 c-4 a^6 c^2+6 a^5 b c^2+9 a^4 b^2 c^2-20 a^3 b^3 c^2+9 a^2 b^4 c^2+6 a b^5 c^2-4 b^6 c^2-10 a^4 b c^3+8 a^3 b^2 c^3+8 a^2 b^3 c^3-10 a b^4 c^3+6 a^4 c^4-2 a^3 b c^4-11 a^2 b^2 c^4-2 a b^3 c^4+6 b^4 c^4+6 a^2 b c^5+6 a b^2 c^5-4 a^2 c^6-4 b^2 c^6+c^8) (a^8-4 a^6 b^2+6 a^4 b^4-4 a^2 b^6+b^8-4 a^7 c+4 a^6 b c+6 a^5 b^2 c-10 a^4 b^3 c-2 a^3 b^4 c+6 a^2 b^5 c+6 a^6 c^2-14 a^5 b c^2+9 a^4 b^2 c^2+8 a^3 b^3 c^2-11 a^2 b^4 c^2+6 a b^5 c^2-4 b^6 c^2+4 a^5 c^3+10 a^4 b c^3-20 a^3 b^2 c^3+8 a^2 b^3 c^3-2 a b^4 c^3-14 a^4 c^4+10 a^3 b c^4+9 a^2 b^2 c^4-10 a b^3 c^4+6 b^4 c^4+4 a^3 c^5-14 a^2 b c^5+6 a b^2 c^5+6 a^2 c^6+4 a b c^6-4 b^2 c^6-4 a c^7+c^8): : ,SEARCH=-9.930822436108384 tT8=(a^2+2 a b+b^2+2 a c-7 b c+c^2) (a^6-12 a^5 b-3 a^4 b^2+20 a^3 b^3-3 a^2 b^4-12 a b^5+b^6+4 a^5 c+20 a^4 b c-14 a^3 b^2 c-14 a^2 b^3 c+20 a b^4 c+4 b^5 c+5 a^4 c^2-34 a^3 b c^2+57 a^2 b^2 c^2-34 a b^3 c^2+5 b^4 c^2-18 a^2 b c^3-18 a b^2 c^3-5 a^2 c^4+44 a b c^4-5 b^2 c^4-4 a c^5-4 b c^5-c^6) (a^6+4 a^5 b+5 a^4 b^2-5 a^2 b^4-4 a b^5-b^6-12 a^5 c+20 a^4 b c-34 a^3 b^2 c-18 a^2 b^3 c+44 a b^4 c-4 b^5 c-3 a^4 c^2-14 a^3 b c^2+57 a^2 b^2 c^2-18 a b^3 c^2-5 b^4 c^2+20 a^3 c^3-14 a^2 b c^3-34 a b^2 c^3-3 a^2 c^4+20 a b c^4+5 b^2 c^4-12 a c^5+4 b c^5+c^6): : ,SEARCH=38.44029689817361 tT9=(a^2-b^2+4 b c-c^2) (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-3 a^5 c+6 a^4 b c-6 a^3 b^2 c-4 a^2 b^3 c+9 a b^4 c-2 b^5 c-a^4 c^2-4 a^3 b c^2+10 a^2 b^2 c^2-4 a b^3 c^2-b^4 c^2+6 a^3 c^3-4 a^2 b c^3-6 a b^2 c^3+4 b^3 c^3-a^2 c^4+6 a b c^4-b^2 c^4-3 a c^5-2 b c^5+c^6) (a^6-3 a^5 b-a^4 b^2+6 a^3 b^3-a^2 b^4-3 a b^5+b^6-2 a^5 c+6 a^4 b c-4 a^3 b^2 c-4 a^2 b^3 c+6 a b^4 c-2 b^5 c-a^4 c^2-6 a^3 b c^2+10 a^2 b^2 c^2-6 a b^3 c^2-b^4 c^2+4 a^3 c^3-4 a^2 b c^3-4 a b^2 c^3+4 b^3 c^3-a^2 c^4+9 a b c^4-b^2 c^4-2 a c^5-2 b c^5+c^6): : ,SEARCH=10.71081597327241 *** Collinear points on K311
 X2, X7, X908 X2, X8, X519 X2, X80, X6224 X2, X903, X30578 X2, X8046, T4 X2, X34234, T3 X2, T0, T7 X2, T1, T6 X2, T2, T5 X2, T8, tT6 X2, T9, tT5 X7, X8, X320 X7, X80, T3 X7, X519, T5 X7, X903, X6224 X7, X8046, T7 X7, X30578, T1 X7, T4, tT6 X8, X80, X30578 X8, X903, T6 X8, X908, tT5 X8, X6224, X34234 X8, X8046, T8 X8, T0, T4 X8, T3, tT9 X8, T7, tT3 X80, X519, X908 X80, X903, T4 X80, T1, T8 X80, T2, T9 X80, T5, T6 X320, X519, X903 X320, X908, X34234 X320, X6224, T0 X320, X8046, X30578 X320, T1, T2 X320, T3, tT3 X320, T4, tT4 X320, T5, tT5 X320, T6, tT6 X320, T7, tT7 X320, T8, tT8 X320, T9, tT9 X519, X6224, X8046 X519, X30578, tT6 X519, X34234, T9 X519, T0, T3 X519, T4, tT8 X519, T7, tT4 X903, X908, T2 X903, X34234, T7 X903, T3, tT5 X908, X6224, tT3 X908, X8046, T6 X908, X30578, T0 X908, T4, tT7 X908, T8, tT4 X6224, X30578, tT4 X6224, T1, tT5 X6224, T2, tT6 X6224, T5, tT9 X6224, T6, tT8 X8046, T2, T3 X30578, X34234, T5 X30578, T3, tT7 X30578, T9, tT3 X34234, T1, T4 T0, T5, T8 T0, T6, T9 T1, T7, tT9 T2, T7, tT8 T3, T6, tT4 T4, T5, tT3 T7, tT5, tT6 T8, T9, tT7
 Note : X and tX are isotomic conjugates hence collinear with X(320).