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X(2), X(187), X(353), X(538), X(1340), X(1341), X(3117), X(5638), X(5639), X(8289), X(11654), X(11655)
other points below
K891 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. K891 is a Psi-cubic as in Table 60.
K891 is an isogonal pK with pivot X(538) in the triangle T = X(2)X(187)X(353) and it must contain the in/excenters of this triangle which are X(1340), X(1341), X(5638), X(5639).
K891 meets its real asymptote again at X = Psi(X3117), on the circumcircle of T, the common tangential of X(2), X(187), X(353), X(538).
X = a^2 (2 a^2 b^2+2 a^2 c^2-b^2 c^2) / (a^4-b^2c^2) : : , SEARCH = -0.0204298459248882.
The singular focus F is the antipode of X on (C), SEARCH = 0.509000028220818.
K891 also contains E = Psi(X8289), on the lines X(2)X(187), X(353)X(538), etc.
E = 4 a^8-3 a^4 b^4-2 a^2 b^6-3 a^4 b^2 c^2-6 a^2 b^4 c^2-3 a^4 c^4-6 a^2 b^2 c^4-6 b^4 c^4-2 a^2 c^6 : : , SEARCH = 6.76103032226340.
X and E are now X(11654) and X(11655) in ETC (2017-01-11).