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X(2), X(3), X(1340), X(1341), X(5657), X(20481), X(22712), X(47044), X(47045), X(47046)

infinite points of K003

vertices of the Thomson triangle Q1Q2Q3

points of nK0(X6, X7736) on (O) i.e. antipodes of points (apart A, B, C) of K243 on (O)

images of A, B, C under h(O, 1/3) = Thomson-isogonal conjugates of the midpoints of ABC

Geometric properties :

K1264 is the homothetic of K028 under h(O, 1/3). Its Thomson-isogonal transform is K1265.

K1264 is a nodal stelloid with node G and radial center X(5054).

The nodal tangents are parallel to the asymptotes of the Jerabek hyperbola.

The asymptotes are parallel to those of the McCay cubic K003. The two cubics meet again at six finite points that lie on the rectangular hyperbola (H) passing through X3, X6, X376, X1992, X2574, X2575, etc.

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Two transformations analogous to the Cundy-Parry transformations of CL037 are related to K1264.

For any M on (O) and its isogonal conjugate M* on (L∞), the point OM /\ GM* lies on K1264.

Consequently, for any M on (L∞) and its isogonal conjugate M* on (O), the point GM /\ OM* lies on K1264.

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K1264 shares lots of properties with K028 such as :

• every circle passing through O and G meets K1264 again at the vertices of an equilateral triangle.

• the circle with center O passing through G meets K1264 again at the vertices of a square.

• the circle with center G passing through O meets K1264 again at the vertices of a regular pentagon.

• every equilateral triangle with center X(5054) and sidelines parallel to the asymptotes meets K1264 again at six points on a same circle whose center lies on the line through X(526) and X(5054).