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X(1), X(1936), X(1937), X(2635), X(2659), X(2660), X(23691), X(23692), X(23693), X(23694), X(23695), X(23707)

X(23707) = X(2635)* = isogonal conjugate of X(2635)

Geometric properties :

K1076 is an isogonal conico-pivotal nodal cubic with node X(1). See Table 69.

It is the locus of M such that M and its isogonal conjugate M* are conjugated in the circle (C) with center X(663) passing through X(1), X(15), X(16), X(36), X(3465), X(4040), X(5526), X(5529).

K1076 is also the X(1)-Hirst inverse of the circum-conic with perspector X(652) passing through X(i) for i = 1, 3, 29, 77, 78, 102, 219, 282, 283, 284, 296, 332, 945, 947, 949, 951, 1036, 1037, 1057, 1059, 1065, 1067, 1069, 1433, 1794, 1795, 1807, 2066, 2338, 2359, 2656, etc.

K1076 is also the X(1)-line conjugate of the bicevian conic C(X1, X653) passing through X(i) for i = 1, 65, 207, 1108, 1148, 2294, 2331, 2658, etc.

These properties are easily adapted for any cK0(#X1, P) hence with P on the antiorthic axis, the trilinear polar of X(1).

***

More generally, any cK(#F, P) is a cK0 if and only if its root P lies on the trilinear polar of F or, equivalently, F lies on circum-conic with perspector P.

In this case, the cubic is

• the F-Hirst conjugate of the circum-conic with perspector P,

• the F-line conjugate of the bicevian conic C(F, P*) where P* is the image of P under the isoconjugation with fixed point F.