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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.

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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

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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).