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K887

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X(2), X(6), X(23), X(111), X(182), X(187), X(353), X(381), X(6032), X(9140), X(10546), X(11645), X(11646), X(11647), X(11648)

other points below

K887 is a circular cubic invariant under the involution Psi described in the page K018 and in the paper "Orthocorrespondence and Orthopivotal Cubics", §5. See also the analogous focal cubic K508. K887 is a Psi-cubic as in Table 60.

Its real point at infinity S is that of the lines X(2)X(10546), X(23)X(9140), X(182)X(381), X(353)X(6032), etc. S is now X(11645) in ETC (2017-01-09).

K887 also contains the following simple points :

• P = Psi(X6032) = a^6-a^4 b^2-b^6-a^4 c^2+a^2 b^2 c^2+b^4 c^2+b^2 c^4-c^6 : : , on the lines X(2)X(353), X(6)X(381), X(187)S, SEARCH = 3.45646842875237.

• P1 = a^6+2 a^4 b^2+5 a^2 b^4-2 b^6+2 a^4 c^2-15 a^2 b^2 c^2+2 b^4 c^2+5 a^2 c^4+2 b^2 c^4-2 c^6 : : , on the lines X(2)X(6), X(111)S, SEARCH = -4.99700070759363.

• Q1 = a^4-2 a^2 b^2-2 b^4-2 a^2 c^2+4 b^2 c^2-2 c^4 : : , on the lines X(2)X(111), X(182)P1, X(6)S, SEARCH = 2.32425284660873.

P = X(11646), P1 = X(11647), Q1 = X(11648) in ETC now (2014-01-10).

K887 is the isogonal pK with pivot S with respect to the triangle T with vertices X(2), X(6), X(111).

It follows that K887 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. See the analogous cubics K881, K886.

The singular focus F is X(14699).