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X(2), X(6), X(111), X(262), X(353), X(524), X(11654), X(11673), X(43535), X(44557)

see points below

Geometric properties :

K1308 is a circular cubic, the member K((4 sin²ω - 1) / 6) of the pencil mentioned in K1156. See also K1306 and K1307.

The singular focus is F such that OF = (4 cos²ω / 3) OX(111), (vectors).

F = a^2 (4 a^10 b^2-9 a^8 b^4+2 a^6 b^6+10 a^4 b^8-6 a^2 b^10-b^12+4 a^10 c^2-31 a^8 b^2 c^2+38 a^6 b^4 c^2-20 a^4 b^6 c^2+35 a^2 b^8 c^2+2 b^10 c^2-9 a^8 c^4+38 a^6 b^2 c^4-33 a^4 b^4 c^4-18 a^2 b^6 c^4-b^8 c^4+2 a^6 c^6-20 a^4 b^2 c^6-18 a^2 b^4 c^6-8 b^6 c^6+10 a^4 c^8+35 a^2 b^2 c^8-b^4 c^8-6 a^2 c^10+2 b^2 c^10-c^12) : : , SEARCH = 0.0829896361821562.

Other points on K1308

MaMbMc is the medial triangle. The line GK meets the sidelines of ABC at U, V, W. The cubic K1308 meets these sidelines at A', B', C' such that A' is the homothetic of U under h(Ma, -(1 + 4 cos 2ω) / 3), B' and C' cyclically.

P1 = -2 a^4+5 a^2 b^2+b^4+5 a^2 c^2-b^2 c^2+c^4 : : , SEARCH = 2.46612210147755, on the lines {2,111}, {524,44557}.

P2 = a^2 (2 a^6 b^2-4 a^4 b^4+a^2 b^6+b^8+2 a^6 c^2-a^4 b^2 c^2+2 a^2 b^4 c^2-b^6 c^2-4 a^4 c^4+2 a^2 b^2 c^4-b^4 c^4+a^2 c^6-b^2 c^6+c^8) : : , SEARCH = 1.01012254383107, on the lines {6,111}, {262,43535}, {11654,44557}.

P3 = a^2 (2 a^4-2 a^2 b^2+2 b^4-5 a^2 c^2-5 b^2 c^2-c^4) (a^2 b^2-b^4+a^2 c^2+3 b^2 c^2-c^4) (2 a^4-5 a^2 b^2-b^4-2 a^2 c^2-5 b^2 c^2+2 c^4) : : , SEARCH = 6.30632771194571, on the lines {111,353}, {262,524}.