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x (y^2 – z^2) / (–a + b + c) = 0

X(2), X(7), X(8), X(145), X(4373), X(7048), X(7057), X(8051), X(8055), X(24313), X(24314)

vertices of the intouch triangle

vertices of the antimedial triangle

Geometric properties :

K1078 is at the same time

• the anticomplement of K1077 = pK(X9, X2).

• the image of K170 under the symbolic substitution SS{a -> √a}.

• the isogonal transform of pK(X32, X56).

• the barycentric quotient of K1079 = pK(X6, X57) by X(1). In other words, the barycentric equation of K1078 is the trilinear equation of K1079.

K1078 and pK(X8, X8) share the same points at infinity and meet again at six finite points on the circum-conic with perspector X(522). These are A, B, C, X(2) counted twice and X(8). This conic also contains X(i) for these i : 29, 85, 92, 178, 189, 257, 312, 333, 1121, 1220, 1311, 1952, 2090, 2399, 2988, 2994, etc.

All the cubics in {K201, K365, K747, K748, K761, K1077, K1078, K1079, K1082} are anharmonically equivalent.


Remarkable polar conics PC(P) of P on K1078:

• PC(X8) is the Feuerbach hyperbola (F).

• PC(X2) is the bicevian conic (C) = C(D1, D2) where D1, D2 are the common points of the line X(2)X(7) and the Steiner ellipse (S). The barycentric product of D1, D2 is X(664), the center of the diagonal hyperbola mentioned in page K1077.