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X(1), X(56), X(266), X(1420), X(1616), X(2137), X(3445), X(7963), X(61635)

isogonal conjugates of X(8834), X(12643), X(12646), X(39124)

vertices of the cevian triangle of X(1420)

vertices of the anticevian triangle of X(266)

vertices of the circumcevian triangle of X(56)

further details below

Geometric properties :

K1427 is the isogonal transform of K201 = pK(X9, X145), also its barycentric product by X(57).

K1426 and K1427 generate a pencil of pKs with same pole X(604), pivot on the line X(1), X(3) and isopivot on the excentral Jerabek hyperbola, the circumconic with perspector X(649). Every pK passes through A, B, C, X(1), X(56), X(266) and the vertices of the anticevian triangle of X(266). Recall that these four latter points are the square roots of X(604).

Apart K1426 and K1427, this pencil contains K632 = pK(X604, X1), pK(X604, X3), K1428 = pK(X604, X65).

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.

***

Points on K1427 :

K1427 passes through :

• the vertices A', B', C' of the cevian triangle of X(1420) and the tangents at these points concur at X, on the line {1,2137), the tangential of the isopivot X(3445). See below.

• the vertices Pa, Pb, Pc of the anticevian triangle of X(266) and the tangents at these points concur at X(1420), the tangential of X(266).

• the vertices Oa, Ob, Oc of the circumcevian triangle of X(56) and the tangents at these points concur at X1616).

X(56) is a point of inflexion on K1427 . The polar conic of X(56) splits into two lines, namely :

• the inflexional tangent (T) which is the trilinear polar of X(109), passing through {6, 41, 48, 56, 73, 172, 198, 604, 1055, 1193, 1400, 1404, 1405, 1428, 1450, 1451, 1458, etc}.

• the harmonic polar (H) which is the trilinear polar of X(57), passing through {513, 663, 855, 1149, 1279, 1284, 1319, 1455, 1456, 1457, 1458, etc}.

K1427 and (H) meet at Ta, Tb, Tc on the internal bisectors of ABC and the tangents at these points obviously concur at X(56).

X = a^2 (a+b-3 c) (a+b-c) (a-3 b+c) (a-b+c) (a^4+4 a^3 b+6 a^2 b^2+4 a b^3+b^4+4 a^3 c-68 a^2 b c+44 a b^2 c-12 b^3 c+6 a^2 c^2+44 a b c^2-26 b^2 c^2+4 a c^3-12 b c^3+c^4) : : , SEARCH = 2.31084616494648.