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X(6), X(1640), X(1641), X(1642), X(1643), X(9171), X(66352), X(66355), X(66356), X(66357), X(66358), X(66359), X(66360), X(66361), X(66362), X(66363), X(66364)

infinite points of the sidelines of ABC

other points below

Geometric properties :

K1382 is the locus of the tripolar centroids of points on the circum-circle. See the general case in CL045. In other words, if a line passing through X(6) meets the sidelines of ABC at three points, then the isobarycenter of these points lies on K1382.

In particular, if KaKbKc is the cevian triangle of X(6), the homothetic A' of Ka under h(A,1/3) lies on K1382, B' and C' likewise.

K1382 is an acnodal cubic with node X(6), an isolated point on the curve.

K1382 has three real asymptotes parallel to the sidelines of ABC. They are their images under the homothety h(X6, 1/3).

K1382 meets these sidelines again at six points on a same circle, namely the (green dotted) Dao-symmedial circle, with center X(5092), square radius 3R² (3 - tan²ω) / 16. See X(5092) in ETC for a description.

K1382 is tritangent to the Steiner inellipse at three points S1, S2, S3 which lie on the Thomson cubic K002 and also on the sidelines of the Thomson triangle. The normals to the ellipse at X(115) and at S1, S2, S3 concur at X(6) hence these four points lie on the (brown dotted) Joachimstahl rectangular hyperbola which passes through {2, 4, 6, 39, 115, 1640, 3413, 3414}. It is homothetic to the Kiepert hyperbola. The orthocenter of S1S2S3 is H and its circumcircle is the circle with center X(51737), the midpoint of X(6), X376), passing through X(2482), the reflection of X(115) in X(2).

K1382 meets the Steiner ellipse at six points which lie on two nK0s with same pole X(385), roots R1 and R2, and also on two nK0s with same root X(385), poles Ω1 and Ω2.

These points are :

R1 = -a^6-2 a^4 b^2-2 a^4 c^2+a^2 b^2 c^2+b^4 c^2+a^2 c^4+2 b^2 c^4 : : , SEARCH = -13.1827605320818

R2 = a^6+2 a^4 b^2-a^2 b^4+2 a^4 c^2-a^2 b^2 c^2-2 b^4 c^2-b^2 c^4 : : , SEARCH = -9.17163874948061

Ω1 = -2 a^6-a^4 b^2-a^4 c^2+2 a^2 b^2 c^2+2 b^4 c^2-a^2 c^4+b^2 c^4 : : , SEARCH = -4.08936512720747

Ω2 = 2 a^6+a^4 b^2+a^2 b^4+a^4 c^2-2 a^2 b^2 c^2-b^4 c^2-2 b^2 c^4 : : , SEARCH = -6.09492601850808

Remark : the lines R1R2 and Ω1Ω2, R1Ω1 and R2Ω2, R1Ω2 and R2Ω1 pass through X(523), X(385), X(6) respectively.