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∑ b c x (y + z) (c y - b z) = 0

X(1), X(4), X(8), X(21147), X(65606), X(65607)

infinite points of the internal bisectors

vertices of the Nagel triangle Q1Q2Q3

vertices of the cevian triangle A'B'C' of X(75)

foci of the inconic (C) with center X(10) and perspector X(75)

Geometric properties :

K1366 is the locus of pivots of pKs meeting the circumcircle again at the vertices of the Nagel triangle. The locus of the poles is psK(X560, X1, X6). See K692.

The tangents to K1366 at Q1, Q2, Q3 concur at X(21150) which is the Lemoine point of the Nagel triangle. It follows that this cubic is also a psK in this triangle hence is must meet its sidelines again at the vertices R1, R2, R3 of the cevian triangle of a point Z whose isogonal conjugate Z' in Q1Q2Q3 is the intersection of the lines {3,902}, {40,78}. Note that (C) is inscribed in both triangles ABC and Q1Q2Q3 with contacts A', B', C' and R1, R2, R3 respectively.

Z = b c (3 a^5-3 a^4 b+4 a^3 b^2+6 a^2 b^3-3 a b^4+b^5-3 a^4 c+4 a^3 b c-6 a^2 b^2 c+b^4 c+4 a^3 c^2-6 a^2 b c^2+6 a b^2 c^2-2 b^3 c^2+6 a^2 c^3-2 b^2 c^3-3 a c^4+b c^4+c^5) : : , SEARCH = 1.73706197493950.

Z' = a^2 (a^5+2 a^2 b^3-a b^4-2 b^5-2 a^2 b^2 c+2 b^4 c-2 a^2 b c^2+6 a b^2 c^2-4 b^3 c^2+2 a^2 c^3-4 b^2 c^3-a c^4+2 b c^4-2 c^5) : : , SEARCH = 10.7470697305612.

P = (a^4-2 a^2 b^2+b^4+2 a^2 b c+2 a b^2 c-2 a b c^2-c^4) (a^4-b^4+2 a^2 b c-2 a b^2 c-2 a^2 c^2+2 a b c^2+c^4) (a^8-2 a^7 b+2 a^5 b^3-2 a^4 b^4+2 a^3 b^5-2 a b^7+b^8-2 a^7 c-4 a^5 b^2 c+2 a^4 b^3 c+6 a^3 b^4 c-4 a^2 b^5 c+2 b^7 c-4 a^5 b c^2+8 a^4 b^2 c^2-8 a^3 b^3 c^2+4 a b^5 c^2+2 a^5 c^3+2 a^4 b c^3-8 a^3 b^2 c^3+8 a^2 b^3 c^3-2 a b^4 c^3-2 b^5 c^3-2 a^4 c^4+6 a^3 b c^4-2 a b^3 c^4-2 b^4 c^4+2 a^3 c^5-4 a^2 b c^5+4 a b^2 c^5-2 b^3 c^5-2 a c^7+2 b c^7+c^8) : : is another point on K1366, SEARCH = 2.18900432432982.

The point S = a (a-2 b-2 c) (a^2-b^2+4 b c-c^2) : : = X(40587) is the only point whose polar conic is a circle. It lies on the lines {2,1000}, {8,442}, {9,374}, {10,1482}, {100,2320}, {119,2886}, {142,519}, etc.

Z, Z', P are now X(65604), X(65605), X(65606) in ETC.