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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 :

  1. Let A1B1C1 be the first Brocard triangle and M a variable point. The lines MA1, MB1, MC1 meet BC, CA, AB at Ma, Mb, Mc respectively. These three points are collinear if and only if M lies on K017. They form a triangle perspective to ABC if and only if M lies on K020. This is also true with the third Brocard triangle. See a generalization at CL041.
  2. Let (C) be the circle with center X(1513) which is orthogonal to the pedal triangle of X(2) and X(6). K017 is the locus of point M whose pedal triangle is orthogonal to (C).
K017a

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 asymptotes.

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.