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K886

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X(2), X(3), X(6), X(110), X(111), X(115), X(542), X(6792)

other points below

K886 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. K886 is a Psi-cubic as in Table 60.

Its singular focus is F = X(9178), the antipode of X(3) on the circumcircle of the triangle T with vertices X(2), X(6), X(111).

The real asymptote is the line X(3), X(542) hence parallel to the Fermat axis.

K886 meets the circumcircle again at two points P, Q on the parallel at X(115) to the Euler line. The points Psi(P), Psi(Q) lie on the cubic, on the Brocard circle, on the line X(115), X(2502).

K886 is the isogonal pK with pivot X(542) with respect to T. Its isopivot is X(3) whose polar conic passes through X(2), X(3), X(6), X(111), X(542), etc, hence K886 is tangent at X(2) to the Euler line.

It follows that K886 must contain the in/excenters of T which are the intersections of the parallels at X(6) to the asymptotes of the Jerabek hyperbola and the parallels at X(2) to the asymptotes of the Kiepert hyperbola. These two latter parallels are the axes of the Steiner inellipse.

In particular, the incenter Io of T is X(2)X(3413) /\ X(6)X(2575) with SEARCH = 1.70769886050969.

These four points also lie on K881 and the two cubics have five remaining common points namely X(2), X(6), X(111) and the circular points at infinity.

Note that X(3), X(110), X(115), X(6792) share the same tangential S which is the Psi-image of the third point of the cubic on the line X(3)X(110).