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X(3), X(74), X(186), X(5663), X(46423)

isogonal conjugate of X(14989)

inverses in the circumcircle of the four foci of every inconic with center on the Euler line

Geometric properties :

K1394 is the isogonal transform of K530 and also the inverse of K187 in the circumcircle. See Table 73 for properties and other related cubics.

It is a focal cubic with singular focus X(186).

The polar conic (C) of X(186) is the circle passing through X(3), X(74), X(107), X(186). See the related K114.

K1394 and K187 meet at A, B, C, X(3), X(74), the circular points at infinity and two points P1, P2 on the line {3, 64}, on the rectangular circum-hyperbola (H) passing through X(2071), X(2693). They are inverse in the circumcircle and symmetric about X(11589).

The orthic line (L) passes through {3, 74, 110, 156, 246, 399, 1511, 1614, 2972, 3470, 5191, 5609, 5663, 5876, 5944, 6090, 6241, etc} hence the real asymptote is its homothetic under h(X186, 2).

It follows that K1394 is the locus of contacts of tangents drawn through X(186) to the circles passing through X(3) and X(74).

On the other hand, if Z is a point on (L), the line through X(186) and Z meets the circle with center Z, orthogonal to the circle with diameter X(3)X(74), at two points of K1394.

K1394 is an isoptic cubic. It is the locus of M such that the directed angles (MX3, MX5663) and (MX186, MX4) are equal.