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K882

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X(2), X(13), X(14), X(15), X(16), X(30), X(184), X(186), X(3165), X(3166)

P, Q see below

K882 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. K882 is a Fermat Psi-cubic as in Table 60 where a generalization is given.

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

The line (L) passing through X(6), X(186) meets the circle (C) passing through X(2), X(111), X(184) at two points P, Q lying on K882 which is an isogonal pK in the triangle GPQ with pivot X(30). It follows that K882 contains the in/excenters of GPQ which are the common points of the axes of the Steiner inellipse and (H).

The singular focus F is the antipode of X(184) on (C) with SEARCH = -12.9586166793637.

F = a^2 (b^2-c^2) (a^8-5 a^4 b^4+6 a^2 b^6-2 b^8-a^4 b^2 c^2+b^6 c^2-5 a^4 c^4+6 a^2 c^6+b^2 c^6-2 c^8) : : .