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X(2), X(3), X(4), X(6), X(9), X(57), X(104), X(572), X(958), X(961), X(1150), X(1766), X(13478)

X(45998) → X(46000), X(46009) → X(46013)

X(46010) = isogonal conjugate of X(5739), the tangential of G

Geometric properties :

K002 and K384 are two isogonal cubics which generate a pencil of circum-cubics stable under isogonal conjugation.

This pencil contains K387, its isogonal transform, two decomposed cubics namely the union of the Jerabek hyperbola and the line X(2)X(7)X(9)X(57), the union of the Euler line and the circum-conic passing through X(6), X(9), X(57).

K1245 and its isogonal transform K1246, K1247 and its isogonal transform K1248 are two other pairs of remarkable members of the pencil. See Table 30.

K1245 is invariant under the involution M -> GM /\ KM* , where M* is the isogonal conjugate of M. This remains true for every cubic of the pencil.

Points on K1245 :

Q1 = a (a+b-c) (a-b+c) (a^6+a^5 b-2 a^4 b^2-2 a^3 b^3+a^2 b^4+a b^5+a^5 c-2 a^4 b c+2 a^3 b^2 c-3 a b^4 c+2 b^5 c-2 a^4 c^2+2 a^3 b c^2-2 a^2 b^2 c^2+2 a b^3 c^2-2 a^3 c^3+2 a b^2 c^3-4 b^3 c^3+a^2 c^4-3 a b c^4+a c^5+2 b c^5) : : , SEARCH = 0.665363953129877, on the lines {X2,X1804}, {X4,X11}, {X6,X57}, {X28,X1035}

Q2 = (a^2+a b+b^2+a c+b c) (a^2+a b+a c+b c+c^2) (a^4-b^4+2 a^2 b c+2 a b^2 c+2 a b c^2+2 b^2 c^2-c^4) : : , SEARCH = 1.68752899703056, on the lines {X4,X572}, {X57,X1150}

Q3 = a (a^3-a b^2+2 a b c-2 b^2 c-a c^2-2 b c^2) (a^3+a^2 b+a b^2+b^3-a^2 c+2 a b c-b^2 c-a c^2-b c^2+c^3) (a^3-a^2 b-a b^2+b^3+a^2 c+2 a b c-b^2 c+a c^2-b c^2+c^3) : : , SEARCH = 4.58185805963473, on the lines {X2,X104}, {X6,X517}

Q4 = (3 a^3-a^2 b-3 a b^2+b^3-a^2 c+2 a b c-b^2 c-3 a c^2-b c^2+c^3) (a^3+a^2 b+a b^2+b^3-a^2 c+4 a b c-b^2 c-a c^2-b c^2+c^3) (a^3-a^2 b-a b^2+b^3+a^2 c+4 a b c-b^2 c+a c^2-b c^2+c^3) : : , SEARCH = 5.13269004665205, on the lines {X57,X1766}, {X104,X631}

Q5 = a (a^3+a^2 b-a b^2-b^3+a^2 c-2 a b c+b^2 c-a c^2+b c^2-c^3) (a^4-2 a^2 b^2+b^4-a^3 c-a^2 b c-a b^2 c-b^3 c-a^2 c^2-4 a b c^2-b^2 c^2+a c^3+b c^3) (a^4-a^3 b-a^2 b^2+a b^3-a^2 b c-4 a b^2 c+b^3 c-2 a^2 c^2-a b c^2-b^2 c^2-b c^3+c^4) : : , SEARCH = 12.9431151596788, on the lines {X4,X958}, {X57,X572}

Q6 = tangential of G = isogonal conjugate of X5739 = a^2 (a^3+a^2 b+a b^2+b^3+a^2 c+2 a b c+b^2 c-a c^2-b c^2-c^3) (a^3+a^2 b-a b^2-b^3+a^2 c+2 a b c-b^2 c+a c^2+b c^2+c^3) : : , SEARCH = -0.0118925450449533, on the lines {X2,X1444}, {X3,X37}, {X6,X1437}, {X25,X1333}, {X28,X393}, {X42,X48}, {X56,X1880}