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K874

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X(2), X(4), X(13), X(14), X(125), X(542)

two imaginary points S1, S2 on the Kiepert hyperbola, the orthocentroidal circle, the line X(115)X(125)

foci of the orthic inconic = inconic with center X(6)

See Table 59 for other similar cubics and a generalization.

K874 is the locus of contacts of the tangents drawn through X(2) to the circles passing through the Fermat points X(13), X(14).

The singular focus is X(2) and the real asymptote is the parallel at X(4) to the Fermat axis.

The polar conic (C) of X(2) is the circle called the Hutson-Parry circle (HP) passing through X(2), X(13), X(14), X(111), X(476), X(5466), X(5640), X(6032), X(6792), X(7698) with center X(8371). (HP) is the Psi-transform of the Fermat axis.

K874a

K874 is a Psi-cubic as in Table 60.

In particular, Psi swaps :

• the polar conic (C) of X(2) and the Fermat axis,

• the Fermat points X(13), X(14),

• X(4) and X(125),

• the real asymptote and the circle passing through X(2), X(125), X(373) tangent at X(2) to the Euler line,

• the foci K1, K2 of the orthic inconic,

• the axes of this latter conic into the circles passing through X(2), X(111) and X(1113), X(1114) respectively,

• the two imaginary points S1, S2 above.

See the analogous K958.