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K884a

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X(2), X(4), X(13), X(14), X(15), X(16), X(30), X(113), X(125), X(5667), X(9730)

P, Q see below

K884 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. K884 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(381), X(395), X(396), X(523), X(3018).

(C) is the circle passing through X(2), X(107), X(111), X(125), X(468), X(1560), X(1637) and the singular focus F which is the antipode of X(125).

F = (b^2-c^2) (-4 a^10+7 a^8 b^2-a^6 b^4-2 a^4 b^6-a^2 b^8+b^10+7 a^8 c^2-16 a^6 b^2 c^2+7 a^4 b^4 c^2+4 a^2 b^6 c^2-2 b^8 c^2-a^6 c^4+7 a^4 b^2 c^4-8 a^2 b^4 c^4+b^6 c^4-2 a^4 c^6+4 a^2 b^2 c^6+b^4 c^6-a^2 c^8-2 b^2 c^8+c^10) : : , SEARCH = -3.96957685667057.

The real asymptote is the parallel at X(125) to the Euler line.

The line (L) passing through X(4), X(6) meets (C) at two points P, Q on K884 which is the isogonal pK with pivot X(30) with respect to the triangle GPQ.

Hence, K884 must contain the in/excenters of GPQ. These are the common points of (H) and the axes of the Steiner ellipses.

K884