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X(1), X(3), X(8), X(40)

points at infinity of K003

vertices of the Nagel triangle, see K692

vertices of the CircumTangential triangle

other points below

Geometric properties :

K1063 is the McCay cubic of the Nagel triangle i.e. the locus of M such that X(3), M and the isogonal conjugate of M in the Nagel triangle are collinear. See the related K078, K1064.

K1063 is a stelloid with radial center X(5657), the centroid of the Nagel triangle, and three real asymptotes parallel at this latter point to those of K003.

K1063 and K003 meet again at six finite points lying on the rectangular hyperbola (H) passing through X1, X3, X21, X100, X224, X1001, the vertices of the circumcevian triangle of X1, the infinite points of the Feuerbach hyperbola.

K1063 meets the sidelines of the CircumTangential triangle T1T2T2 again at N1, N2, N3 which are the vertices of the pedal triangle of X(8) in T1T2T2. These points N1, N2, N3 lie on the parallels at X(8) to the asymptotes of K003.

K1063 meets the sidelines of the Nagel triangle Q1Q2Q3 again at R1, R2, R3 which are the vertices of the cevian triangle of X(3) in Q1Q2Q3. The tangents to K1063 at Q1, Q2, Q3 and X(3) concur at X(8). The tangents at R1, R2, R3 and X(8) concur at a point P2 on K1063 with SEARCH = 5.32482605950432 and rather complicated barycentrics. P2 = tangential of X(8) = X(21227) is now in ETC (2018-08-22)

Note that the polar conic of X(8) in K1063 is the Jerabek hyperbola of Q1Q2Q3 passing through X3, X8, X513, X517, X901, X1149, etc.

The third point P1 of K1063 on the line X(1)X(8) is simple with 1st barycentric : a (a^2 b+a b^2+a^2 c-b^2 c+a c^2-b c^2-3 a b c), SEARCH = 84.0928483454874, on the lines {9,1050}, {36,1044}, {40,1054}, {56,87}, {106,979}, {171,1191}, etc. P1 = X(21214) is now in ETC (2018-08-20). P1 is the "last" common point of K1063 and K078 which is the McCay cubic of the Thomson triangle.

Other points on K1063 now added to ETC :

X(21228), X(21229), X(21306) = tangential of X(1), (21307) = tangential of X(40).