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K1273

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X(40), X(55), X(57), X(365), X(1436), X(2066), X(2362), X(3197), X(3345), X(5414), X(16232), X(32555), X(32556)

isogonal conjugates of X(46421), X(46422) and CPCC points, see Table 11

vertices of the cevian triangle of X(40)

vertices of the anticevian triangle of X(365)

points of (O) on K1044

Geometric properties :

K1273 is the isogonal transform of K332.

It is a member of the pencil of cubics pK(X31, P on the line IO) that contains K131, K145, K1253 obtained with P = X(171), X(3), X(165) respectively.

K1273 meets the circumcircle (O) at the same points as infinitely many pKs with pole on psK(X9448, X55, X6) and pivot on K710 = psK(X55, X8, X4) = spK(X7, X9). K1044, pK(X3 x X55, X55) and pK(X1253, X9) are three of these cubics.

K1273 meets the line at infinity at the same points as infinitely many pKs. Here are three remarkable examples :

• pK(X6, P1) with P1 = a (a^3 b+a^2 b^2-a b^3-b^4+a^3 c-a^2 b c-a b^2 c+b^3 c+a^2 c^2-a b c^2-a c^3+b c^3-c^4) : : , SEARCH = 9.32808967359447, on the lines {X2, X2262}, {X3, X1442}, {X7, X517}, {X40, X77}, {X46, X1014}, {X100, X326}.

• pK(X1407, P2) with P2 = a (a+b-c)^2 (a-b+c)^2 (a b-b^2+a c-b c-c^2) : : , SEARCH = -0.696117419488586, on the lines {X1, X103}, {X4, X7}, {X40, X77}, {X41, X57}, {X65, X279}, {X81, X2332}, {X169, X651}.

• pK(Ω, X4) with Ω = a (a^2+b^2-c^2) (a^2-b^2+c^2) (a^3 b+a^2 b^2-a b^3-b^4+a^3 c-2 a^2 b c+a b^2 c+a^2 c^2+a b c^2+2 b^2 c^2-a c^3-c^4) : : , SEARCH = 0.119222474134403, on the lines {X19, X3195}, {X25, X1096}, {X31, X2355}, {X34, X1407}, {X42, X1827}.

K1273 meets the Darboux cubic K004 at A, B, C, X(40), X(3345) and the isogonal conjugates of the CPCC points, see Table 11.

K1273 meets K1044 at six points on (O), X(57) and two other points on the line {X6, X57}.