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X(2), X(4), X(5), X(13), X(14), X(542)

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

other points below

See Table 59 for other similar cubics and a generalization.

K872 is a focal cubic with singular focus X(5) whose polar conic is the Lester circle. Its orthic line is the Fermat axis. See also K871.

It is the locus of contacts of the tangents drawn through X(5) to the circles passing through the Fermat points X(13), X(14).

The real asymptote is the homothetic of the Fermat axis under h(O,2), a line passing through X(542), X(1353).

The intersection with this asymptote is X = X(15358), also on the line X(51)X(526).

Y = X(15359) lies on the lines X(5)X(542), X(115)X(125).


K871 and K872 generate a pencil of focal cubics passing through X(2), X(4), X(13), X(14), X(542), S1, S2 and the circular points at infinity.

The singular focus lies on the Euler line and its polar conic is a circle centered on the line X(115)X(125), the perpendicular bisector of X(13), X(14).

Each cubic meets its real asymptote again at X lying on the hyperbola passing through X(2), X(4), X(6), X(523), X(542).

This pencil also contains :

• the decomposed cubic into the line at infinity and the Kiepert hyperbola,

• the decomposed cubic into the Fermat axis and the orthocentroidal circle,

• the axial cubic K873.


We know that K871 and K872 have the same circular polar conic, namely the Lester circle.

More generally, if P and Q are two points harmonically conjugated with respect to X(2), X(4) then the two cubics of the pencil with singular foci P, Q have the same circular polar conic.

Obviously, if P is one of the points X(2), X(4) then P = Q and the two cubics coincide into a cubic tangent at P to the Euler line. See K874 and K875 respectively.