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∑ a^2 (y - z) (x y + x z - 2 y z) = 0

∑ x^2 ((b^2 + 2 c^2) y - (2 b^2 + c^2) z) = 0

X(2), X(6), X(458), X(598), X(4393), X(14608), X(41259)

X(60856) → X(60873)

infinite points of pK(X2, X599)

O1, O2, O3 : points of pK(X32, X7771) and pK(X11003,X6) on the circumcircle

S1, S2, S3 : points of K659 = pK(X2, X6) on the Steiner ellipse

other points below

Geometric properties :

See CL045 for related definitions.

K1359 is the locus of M such that

• the Tripolar Centroid TG(M) of M lies on the trilinear polar of X(598), a line passing through X(351), X(523) hence perpendicular to the Euler line.

• X(6), M and the Tripolar Centroidal Conjugate TCC(M) of M are collinear. See also K185 and K295.

K1359 is also an example of isotomic spK as in CL055. Indeed, let (L) and (L') be two parallel lines passing through X(599) and X(6) respectively. The circum-conic which is the isotomic transform of (L) meets (L') at two points on the cubic.

K1359 is a nodal cubic with node G. The nodal tangents are parallel to the asymptotes of the Kiepert hyperbola hence they are the axes of the Steiner ellipse.

K1359 meets the sideline BC at U, on the parallel at G to the line through A and X(598), equivalently, on the parallel at X(6) to the line through A and X(599).

U = 0 : 2 a^2 + b^2 : 2 a^2 + c^2. V and W are defined likewise.

The third point on the symmedian (AK) is U' = 2 b^2 c^2 : b^2 (b^2+c^2) : c^2 (b^2+c^2). V' and W' are defined likewise.

The tangents at S1, S2, S3 to K1359 concur at S = 3 b^2 c^2 + 2 a^2 (a^2+b^2+c^2) : : , SEARCH = 2.09952020481909, on the lines {2,187}, {6,76}.

Parametrization :

If P = u : v : w ≠ G is a point then the following point Q lies on K1359.

Q = - 2 a^2 + [b^2 (w - u) + c^2 (u - v)] / (v - w) : : .