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X(2), X(3), X(23), X(110), X(182), X(187), X(353), X(9829), X(11645), X(15080)

X(39162), X(39163), X(39164), X(39165) : foci of the Steiner inellipse

vertices of the Grebe triangle

Geometric properties :

Every circular cubic that passes through X(2), X(3), X(110) and the foci of the Steiner inellipse is a focal cubic of the pencil containing K463, K816, K893 and K1286.

The three remaining points on (O) lie on an isogonal pK with pivot on the line GK. K463 and K1286 are the cubics associated with K002 and K102, hence to the Thomson and Grebe triangles.

The singular focus of each cubic lies on the circle (C) passing through {2, 3, 110, 842, 8724, 14649, 14685, 34291, 35911}, with center X(44814).

(C) is the inverse in (O) of the line {23, 110, 323, 511, 842, 1495, 3292, 9970, 10752, 12584, 13402, 13417, 15107, 19140, 23061, 32237, 35265}.

(C) is also the Psi-image of the line {3, 74, 110, 156, 246, 399, 1511, 1614, 2972, 3470, etc}.

The singular focus F = X(51800) of K1286 is the inverse of X(19140) and the Psi-image of X(15080). It is the common tangential of X(2) and X(15080).

K1286 is invariant under the Psi involution (see K018, also "Orthocorrespondence and Orthopivotal Cubics", §5) and under isogonal conjugation with respect to the triangle X(2), X(3), X(110). It is an isogonal nK in this triangle since X(23), X(182), X(15080) are collinear, hence it is the locus of foci of conics inscribed in this triangle with center of the line X(2), X(1495), X(15080).

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More centers on K1286

P1 = X(51798) = 5 a^6-3 a^4 b^2-b^6-3 a^4 c^2-3 a^2 b^2 c^2+3 b^4 c^2+3 b^2 c^4-c^6 : : , SEARCH = 1.80226915622506

P2 = X(51797) = a^2 (4 a^6-4 a^2 b^4-a^2 b^2 c^2-2 b^4 c^2-4 a^2 c^4-2 b^2 c^4): : , SEARCH = -3.96720670910920

P3 = X(51799) = a^2 (4 a^8-4 a^6 b^2+4 a^2 b^6-4 b^8-4 a^6 c^2+3 a^4 b^2 c^2+6 a^2 b^4 c^2+2 b^6 c^2+6 a^2 b^2 c^4+12 b^4 c^4+4 a^2 c^6+2 b^2 c^6-4 c^8): : , SEARCH = 1.07298617312936