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X(1), X(6), X(56), X(57), X(266), X(289), X(1743), X(40151)

isogonal conjugates of X(7028), X(24150), X(24151), X(24152), X(24153), X(24154), X(24155), X(24156), X(24157), X(24158), X(39121)

vertices of the cevian triangle of X(57)

vertices of the anticevian triangle of X(266)

other points below

Geometric properties :

K1426 = pK(X604, X57) is the isogonal transform of K1077 = pK(X9, X2), its barycentric product by X(57) and also the barycentric products X(1) x K365, X(7) x K761.

K1426 meets the line at infinity at the same points as pK(X3, aX1828), pK(X3752, X69). The anticomplement aX(1828) of X(1828) lies on the lines {X2, X1828}, {X20, X145}, {X22, X56}, {X46, X58}, etc.

K1426 meets the circumcircle at the same points Q1, Q2, Q3 as pK(X31, X1743), pK(X32, X56), pK(X1397, X6), pK(X6 x X1743, X1) and actually infinitely many pK(Ω, P) with pivot P on K1426 itself, pole Ω on pK(X32 x X604, X604) and isopivot Q on pK(X32, X56) which is one of these cubics. The tangents at Q1, Q2, Q3 to K1426 are concurrent at X = a^2 (a+b-c) (a-b+c) (a^2-4 a b+3 b^2-4 a c-26 b c+3 c^2) : : , SEARCH = 0.302761114771228, but X is not a point on K1426.

The inconic with perspector X(57) is also inscribed in Q1Q2Q3 and K1426 passes through its contacts R1, R2, R3 with the sidelines of Q1Q2Q3.

K1426 contains the isogonal conjugates Za, Zb, Zc of the vertices of the 2nd Zaniah triangle, see ETC, X(18214). This triangle ZaZbZc is perpective to ABC at X(56), to the cevian triangle of X(57) at X(1743), and to the anticevian triangle of X(266) at the isogonal conjugate of X(24158).

More generally, ZaZbZc is perpective to

• the cevian triangle of every point P on pK(X7366, X269) which contains {X56, X57, X269, X1407, X1420}.

• the anticevian triangle of every point P on K1427 = pK(X604, X1420) which contains {X1, X56, X266, X1420, X1616, X2137}.

In both cases, the locus of the perspector is K1426 itself. Note that X(7366) is the barycentric cube of X(57).

A locus property :

The anticevian triangle of P is perspective to the circumcevian triangle of X(56) if and only if P lies on K1426. The locus of the perspector is K1427 as above.