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X(1), X(3), X(36), X(54), X(1157), X(1687), X(1688), X(2574), X(2575)
excenters and their inverses in the circumcircle
vertices of the circumnormal triangle
X(2574), X(2575) : points at infinity of the Jerabek hyperbola
Q023 is a circular quartic with singular focus X(2070), the inverse of the nine-point center X(5) in the circumcircle. It has two real asymptotes parallel to those of the Jerabek hyperbola intersecting at the point X = a^4 SA (4SA^2 - b^2c^2) : : not mentioned in the current edition of ETC. X is the barycentric product X3 x X323 and lies on many lines e.g. X2, X567 - X3, X49 - X20, X156 - X30, X110 - X36, X215 - X54, X140 - X125, X539 - etc.
O is a node on Q023 with two tangents parallel to the asymptotes of the Jerabek hyperbola.
Q023 is invariant in the inversion with respect to the circumcircle. It meets the circumcircle at A, B, C and the vertices U, V, W of the circumnormal triangle, these six points on the McCay cubic. Obviously, the tangents at these six points pass through O. More generally, the polar line with respect to Q023 of any point on the Lemoine cubic K009 passes through O.
Q023 contains I, the excenters Ia, Ib, Ic and their inverses X(36), Ja, Jb, Jc. Notice that ABC and JaJbJc are perspective at X(35) not on Q023.
Q023 meets the sideline BC at two other points A1, A2 lying on the circle (Ca) passing through A and O, centered at the intersection of BC and the perpendicular bisector of AO. B1, B2 on AC and C1, C2 on AB are defined similarly. Notice that the three circles (Ca), (Cb), (Cc) belong to the same pencil and contain O and X(1157), the inverse of X(54) in the circumcircle.
Q023 is the isogonal transform of Q038, an antigonal circular quintic.
Q023 is closely related to the Lemoine cubic K009 at least in two ways :
See the similar inversible quartic Q098.
Locus properties :