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X(2), X(184), X(186), X(187), X(353), X(1340), X(1341), X(3148), X(5638), X(5639), X(11653)

other points below

K890 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. K890 is a Psi-cubic as in Table 60.

K890 is an isogonal pK with pivot S (the infinite point of the line X184, X3148) in the triangle 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).

S = a^2 (a^6 b^2-a^4 b^4+a^2 b^6-b^8+a^6 c^2-a^2 b^4 c^2-a^4 c^4-a^2 b^2 c^4+2 b^4 c^4+a^2 c^6-c^8) : : , SEARCH = 0.970254678550255.

The singular focus F is complicated with SEARCH = 0.258563183348133.

K890 contains E = Psi(X3148) = a^2 (a^4-2 a^2 b^2+b^4-2 a^2 c^2+4 b^2 c^2+c^4)/(-a^2 b^2+b^4-a^2 c^2+c^4) : : , on the line X(2)X(184), SEARCH = 5.36978913055209. E is now X(11653) in ETC (2017-01-11).

K890 meets the circumcircle again at P1, P2 on the line X(6)X(184) and P3, P4 on the line X(6)X(3148).

Their respective Psi-images Q1, Q2, Q3, Q4 lie on the Brocard circle. Moreover, Q1, Q2 lie on the line X(184)X(574) and Q3, Q4 lie on the line X(3148)X(574).

A lot of other collinearities on K890 can be found e.g. P1, P3, X(5638) โ€“ P1, P4, X(5639) โ€“ P1, Q1, E โ€“ P1, Q2, X(186), etc.