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X(6), X(9), X(19), X(37), X(71), X(198), X(1400), X(1903)

vertices of the cevian triangle of X(9)

other points below

Geometric properties :

We meet this cubic in relation to K1089.

X(37) is point of inflexion whose polar conic splits into the inflexional tangent (T) passing through X(42) and the harmonic polar (P) passing through X(44), X(513). These two lines meet at X(2238) on the Hessian (H) of K1090.

The polar conic of the pivot X(9) is the diagonal rectangular hyperbola (C) passing though X(i) for i in {1, 9, 40, 188, 191, 366, 1045, 1050, 1490, 2136, 2949, 2950, 2951, 3174, etc} and the (always real) square roots Ri of X(213). These are Ro = a Sqrt[a (b+c)] : b Sqrt[b (a+c)] : c Sqrt[(a+b) c] and its harmonic associates Ra, Rb, Rc. Recall that these points Ri share the same tangential, namely X(9).

Other reasonable points :

a (a+b-c) (a-b+c) (b+c) (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4) : : , the tangential of X(1400), on the lines {6,19}, {37,1903}, SEARCH = -0.3965413535569828

a^2 (a-b-c) (a^4-2 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2-3 c^4) (a^4+2 a^2 b^2-3 b^4-2 a^2 c^2+2 b^2 c^2+c^4) : : , on the line {71,198}, SEARCH = 4.65691242585103