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X(514), X(1019), X(1577), X(23594), X(23595), X(23596), X(23597), X(23598), X(23599)

Geometric properties :

K1074 is a conico-pivotal nodal cubic with pivotal conic the parabola with focus X(1054) and directrix the line passing through X(1), X(2).

K1074 is a trident having an asymptote which is the trilinear polar L(X7) of the Gergonne point and also a parabolic asymptote (P) with equation :

∑ sb sc x^2 + [a sa + (b - c)^2] y z = 0, where sa = (– a + b + c)/2 as usual.

L(X7) and (P) meet at X(7658). The tangent to (P) at this point is L(X9311) which contains the three finite points of inflexion of K1074 but one only is real.

Locus properties

1. K1074 is the image under 𝛕_I of the circum-conic C(X514) with perspector X(514). For X on C(X514), X' = 𝛕_I (X) is the intersection of the trilinear polar L(X) of X and the polar L'(X) of X in C(X1). See K1065 for general properties. This transformation is given by :

𝛕_I : X = x : y : z → X' = x^2 (c y - b z) : y^2 (a z - c x) : z^2 (b x - a y).

2. Peter Moses observes that K1074 is the locus of points Z(t) = (b-c) (t a + b + c) (t a + t b + c) (t a + t c + b) : : , where t is a real parameter or infinity.

For example Z(0) = X(1577), Z(1) = X(514), Z(∞) = X(1019). Note that Z(t) and Z(1/t) are swapped under the isoconjugation with fixed point X(514) hence the line Z(t)Z(1/t) is tangent at T(t) to the pivotal parabola. The harmonic conjugate of T(t) in Z(t), Z(1/t) is the third point of K1074 on this line Z(t)Z(1/t).