∑ (a^4 - b^2c^2) x (c^2y^2 + b^2z^2) = 0
∏ (b^2 x - c^2 y) - ∏ (b^2 z - c^2 x) = 0
X(2), X(6), X(99), X(512)
vertices of the first and third Brocard triangles
feet of the trilinear polar of X(385)
X = a^2(b-c) / [(a^4+b^2c^2)(b^2+c^2)-2a^2(b^4+c^4)] : :
X* = X(99)X(512) /\ X(2)X(6)
Z = X(2)X(512) /\ X(6)X(99)
= (b^2+c^2-2a^2) / (a^2b^2+a^2c^2-2b^2c^2)
Z* = X(6)X(512) /\ X(2)X(99)
K017 is an isogonal nK0 with root X(385) meeting the line at infinity at X(512) and two other imaginary points on the Steiner circum-ellipse.
The real asymptote is parallel to the Lemoine axis and meets the cubic at X.
Locus properties :
The polar conic of X(512) passes through X(39), X(187), X(512), X(538) and meets K017 at four points where the tangents are parallel to the real asymptote.
These points are the common points of K017 and the axes of the Steiner ellipses.
The diagonal triangle of these four points is X(2) X(6) X(99).
It follows that K017 is a pivotal cubic in this triangle with pivot X(512), invariant in the isoconjugation that swaps X(512) and X.
The polar conic of X meets K017 at X and X(2), X(6), X(99), X(512). The diagonal triangle of these four latter points is X*, Z, Z*. Hence, K017 is also a pivotal cubic in this diagonal triangle and its pivot is X.