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K885

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X(2), X(5), X(13), X(14), X(15), X(16), X(30), X(265), X(568), X(3448), X(10277)

K885 is a circular cubic invariant under the involution Psi described in the page K018 and in the paper "Orthocorrespondence and Orthopivotal Cubics", ยง5. See also the analogous focal cubic K508. K885 is a Fermat Psi-cubic as in Table 60 where a generalization is given.

The polar conic of X(30) is the rectangular hyperbola (H) passing through X(30), X(395), X(396), X(523), X(547).

(C) is the circle passing through X(2), X(111), X(858), X(925), X(3448) and the singular focus F which is the antipode of X(3448) on (C).

F = (b^2-c^2) (-3 a^10+7 a^8 b^2-6 a^6 b^4+4 a^4 b^6-3 a^2 b^8+b^10+7 a^8 c^2-12 a^6 b^2 c^2+5 a^4 b^4 c^2+4 a^2 b^6 c^2-2 b^8 c^2-6 a^6 c^4+5 a^4 b^2 c^4-7 a^2 b^4 c^4+b^6 c^4+4 a^4 c^6+4 a^2 b^2 c^6+b^4 c^6-3 a^2 c^8-2 b^2 c^8+c^10) : : , SEARCH = -1.40118297908315.

The real asymptote is the parallel at X(3448) to the Euler line.