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X(2), X(3), X(4), X(6), X(7), X(8), X(672), X(1193), X(4259), X(26893)

X(45962) → X(45966), X(45974)

X(45966) = isogonal conjugate of X(16992), the tangential of G

see points below

Geometric properties :

K1244 is the unique circum-cubic passing through X(2), X(3), X(4), X(6), X(7), X(8).

K1244 is invariant under the involution M → GM /\ KM* , where M* is the isogonal conjugate of M.

K1244 is a member of the pencil generated by K386 and K387. This pencil contains two decomposed cubics namely the union of the Jerabek hyperbola and the line X(1)X(2), the union of the Euler line and the circum-conic passing through X(6), X(8).

Points on K1244 :

Q1 = -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 = 3.96972393325751, on the lines {X2,X6}, {X8,X304}, {X22,X1444}, {X75,X1370}, {X76,X2478}

Q2 = a^2 (a+b-c) (a-b+c) (a^3 b^2-a^2 b^3-a b^4+b^5+a^3 b c-a b^3 c+a^3 c^2-b^3 c^2-a^2 c^3-a b c^3-b^2 c^3-a c^4+c^5) : : , SEARCH = -1.74911994295683, on the lines {X1,X185}, {X4,X7}, {X6,X41}, {X25,X222}, {X51,X223}, {X57,X851}, {X77,X511}

Q3 = (a+b-c) (a-b+c) (a^3-a^2 b-a b^2+b^3-2 a b c-a c^2-b c^2) (a^3-a b^2-a^2 c-2 a b c-b^2 c-a c^2+c^3) (a^3 b-3 a^2 b^2+a b^3+b^4+a^3 c-2 a^2 b c-a b^2 c-3 a^2 c^2-a b c^2-2 b^2 c^2+a c^3+c^4) : : , SEARCH = 0.773720146175088, on the line {X3,X7}

Q4 = (a^3-a^2 b-a b^2+b^3+2 a b c-a c^2-b c^2) (a^2 b+2 a b^2-b^3+a^2 c+b^2 c+2 a c^2+b c^2-c^3) (a^3-a b^2-a^2 c+2 a b c-b^2 c-a c^2+c^3) : : , SEARCH = 8.03298273573471, on the line {X3,X8}

Q5 = (a^2 b^2-b^4+a^2 b c+a b^2 c+2 a^2 c^2+a b c^2+b^2 c^2) (2 a^2 b^2+a^2 b c+a b^2 c+a^2 c^2+a b c^2+b^2 c^2-c^4) : : , SEARCH = 1.41234341953170, on the line {X2,X4259} and on the Kiepert hyperbola, the isogonal conjugate of X(5135)

Q6 = (a^3 b^2-a b^4+a^3 b c+a^2 b^2 c-a b^3 c-b^4 c+2 a^3 c^2+2 a^2 b c^2+a b^2 c^2+2 a^2 c^3+a b c^3+b^2 c^3) (2 a^3 b^2+2 a^2 b^3+a^3 b c+2 a^2 b^2 c+a b^3 c+a^3 c^2+a^2 b c^2+a b^2 c^2+b^3 c^2-a b c^3-a c^4-b c^4) : : , SEARCH = -0.0369823132382470, on the line {X2,X26893}