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X(2), X(6), X(13), X(14), X(15), X(16), X(30), X(111), X(1995), X(2549), X(5913)

other points and detailed figure below

K881 is a remarkable circular cubic since it is invariant under at least six involutions.

• Psi described in the page K018 and in the paper "Orthocorrespondence and Orthopivotal Cubics", §5. See also the analogous focal cubic K508. K881 is a Fermat Psi-cubic as in Table 60 where a generalization is given.

• Isogonal conjugation since it is an isogonal pK with respect to the (yellow) triangle X(2)X(6)X(111). The pivot is X(30), the infinite point of the Euler line.

• Four inversions, see below.

The singular focus of K881 is X(5653), the orthocenter of X(3)X(6)X(110), a point on the Jerabek-Thomson hyperbola and on the circle (C) passing through X(2), X(3), X(6), X(111), X(691) and the point X where K881 meets its real asymptote. X is the antipode of X(5653) on (C) and X = Psi(X1995). (C) is the Psi-image of the line passing through X(6), X(110), X(111), X(895), X(1995), X(2493), X(2502), X(2503), X(2854), X(2930), X(3066), X(3124), etc, and its center is X(9175).

X is unlisted in ETC with SEARCH = 6.86770841516032. Its polar conic contains X(2), X(6), X(30), X(111) hence the tangents at these points pass through X.

The polar conic of X(111) contains X(13), X(14), X(15), X(16) hence the tangents at these points pass through X(111).

The polar conic (H) of X(30) is a rectangular hyperbola passing through X(30), X(395), X(396), X(523) whose center Ω is the second intersection of the real asymptote and (C). (H) contains the four centers of anallagmaty Io, Ia, Ib, Ic which are the in/excenters of the triangle X(2)X(6)X(111). This gives the four inversions above. Note that these four points 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. See also K886 and K887.

K881 and K001 meet the line at infinity at the same points, both contain X(13), X(14), X(15), X(16) hence they must meet at two other finite points E1, E2. These two points lie on the line X(74)X(691) hence they are X(30)-Ceva conjugates. They also lie on the (green) conic (𝛄) passing through X(13), X(14), X(15), X(16) whose center is the intersection of the lines X(2)X(6), X(23)X(111), X(110)X(2030), etc, with SEARCH = –1.16553471569476 and with first barycentric coordinate a^2 (4 Sω^2 – 9 b^2c^2).


Additional remarks :

• X lies on the parallel at X(74) to the Brocard axis. X is the image of X(74) under the translation that maps X(3) onto X(6), also the image of X(6) under the translation that maps X(3) onto X(74).

• The pole of the Brocard axis in (𝛄) is X(23) and that of the Fermat axis is X(10989) = reflection of X(23) in X(2).

The pole of the line X(74), X(691) is X(111) hence the tangents at E1, E2 to (𝛄) pass through X(111).