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X(1), X(9), X(57), X(173), X(258), X(1743), X(2136), X(2137), X(8056), X(8078), X(24242)

excenters

A', B', C' : vertices of the cevian triangle of X(57)

Geometric properties :

K1079 is the barycentric product X(1) x K1078. In other words, the barycentric equation of K1078 is the trilinear equation of K1079.

The orthic line of K1079 is the line passing through X(1) and X(3) hence the polar conic PC(P) of any P on this line is a rectangular hyperbola. In particular,

• PC(X57) is the diagonal rectangular hyperbola with center X(934) on (O) that passes through X(1), X(57), X(174), X(223), X(2124), X(3182) and the excenters.

• PC(X1) is the bicevian rectangular hyperbola C(S1, S2) where S1, S2 are the common points of the orthic line and the circum-conic with perspector X(1). PC(X1) contains X(1), X(164), X(173) and its center is X(1054).

On the other hand, PC(X9) is the circum-conic with perspector X(663) that passes through X(i) for these i : 6, 9, 19, 55, 57, 284, 333, 673, 893, 909, 1024, 1174, 1436, 1751, 1945, 2160, 2161, 2164, 2195, 2258, 2259, 2291, 2299, 2316, 2319, 2337, 2339, 2343, 2364, 2432, 2590, 2591, etc. This is the isogonal transform of the line X(2), X(7), X(9), etc.

PC(X2136) is also remarkable since it is the bicevian conic C(X57, X3699) passing through X(9), X(2136), A', B', C'.

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

Additional remarks :

• K1079 contains gX(8078) on the lines {57, 8078}, {173, 1743}, {258, 2136}, etc. It is the X(9)-crossconjugate of X(258), in short 9©258.

More generally, K1079 contains 9©P when P lies on K1079, in particular those (unlisted) of X(173), X(1743), X(2136).

The latter point is the tangential of X(2136), the 6th member of the chain of tangentials X(173), X(1), X(57), X(9), X(2136).

• K1079 contains the X(57)-Ceva conjugate of any of its points, in particular those (unlisted) of X(258), X(2137), X(8056).

The two latter points are the tangentials of X(8056), gX(8078) respectively.