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

S1, S2 defined at the page K019, now X(13414), X(13415) in ETC. See also Table 62.

foci of the orthic inconic or 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 and K1257.

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.