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X(2), X(6), X(111), X(186), X(249), X(524), X(14846), X(41134), X(52699)

see points below

Geometric properties :

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

The singular focus is F = X(52698) such that OF = 2/3 OX(111), (vectors).

F = a^2 (3 a^8-10 a^6 b^2+2 a^4 b^4+10 a^2 b^6-5 b^8-10 a^6 c^2+35 a^4 b^2 c^2-25 a^2 b^4 c^2+20 b^6 c^2+2 a^4 c^4-25 a^2 b^2 c^4-22 b^4 c^4+10 a^2 c^6+20 b^2 c^6-5 c^8) : : , SEARCH = 3.06574888248223.

Other points on K1307

MaMbMc is the medial triangle. The line GK meets the sidelines of ABC at U, V, W. The cubic K1307 meets these sidelines at A', B', C' such that A' is the homothetic of U under h(Ma, -1/3), B' and C' cyclically.

P1 = X(52699) = a^2 (3 a^6-2 a^4 b^2-3 a^2 b^4+2 b^6-2 a^4 c^2+5 a^2 b^2 c^2-b^4 c^2-3 a^2 c^4-b^2 c^4+2 c^6) : : , SEARCH = 0.964459988077677, on the lines {6,111}, {74,182}, {186,249}.

Note that K1307 is tangent at G to the Euler line.