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X(3), X(468), X(511), X(1155)

S1, S2 defined at the page K019. See also Table 62.

foci of the K-ellipse (inellipse with center K when the triangle ABC is acute angle)

The Brocard (third) cubic K019 and the McCay hessian cubic K048 generate a pencil of focal cubics with singular focus on the line GX(110). All these cubics contain the nine following points : X(511), the four foci of the K-ellipse (inconic with center K), two points S1, S2 defined at the page K019 and the circular points at infinity.

This pencil also contains K417 passing through H and the in/excenters.

The only strophoid of the pencil is K418 with node O, where the tangents are parallel to the asymptotes of the circum rectangular hyperbola passing through X(25).

The singular focus F is the midpoint X(6036) of O and X(115), the center of the Kiepert hyperbola.

The real asymptote is the homothetic of the Brocard axis under h(F, 2). K418 meets this asymptote at X on the perpendicular at O to the line OX(115).

K418 is the locus of the intersections of the circles centered on OK passing through O with the diameter which passes through F.

K418 is also invariant under the JS involution described in Table 62.