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X(2), X(3), X(22), X(7712),X(46264)

vertices of the Grebe triangle

infinite points of (K3) = pK(X6, P), see below

points on (O) and (K2) = antipodes of the points on (O) and K006

anti-points, see Table 77

further details below

Geometric properties :

K1249 is an example of non-circular, non-equilateral cubic passing through the anti-points, see Table 77. It is analogous to K727. See also K1250.

K1249 meets (O) at two triples of points, namely :

• the vertices G1, G2, G3 of the Grebe triangle, these points on K102.

• the points Q1, Q2, Q3 on (K2) = nK0(X6, X7735) and on the rectangular hyperbola (H) passing through {3, 20, 74, 2574, 2575} which is the image of the Jerabek hyperbola in the translation that sends H onto O. The antipodes O1, O2, O3 of Q1, Q2, Q3 on (O) lie on the Orthocubic K006.

K1249 meets the line at infinty at three points on (K3) = pK(X6, P) where P = 5 a^6-3 a^2 b^4-2 b^6-2 a^2 b^2 c^2+2 b^4 c^2-3 a^2 c^4+2 b^2 c^4-2 c^6,-2 a^6 : : , SEARCH = 25.9108731053114, on the lines {X3,X2916}, {X6,X30}, {X20,X64}, {X22,X1853}, {X141,X376}, {X154,X1370}, {X159,X2935}. P is X(48905) in ETC.

K1249 meets (K3) again at six finite points that lie on a same circle passing through X(15), X(16) whose center is X(3005). Recall that X(3005) is the intersection of the Lemoine axis and the de Longchamps line.

K1249 meets (H) at O, Q1, Q2, Q3, X(7712) and a sixth point Q which is the Grebe isogonal conjugate of X(32), the tangential of O in K1249, the image of K in the translation above, the midpoint of KP.

Q =3 a^6-a^4 b^2-a^2 b^4-b^6-a^4 c^2-2 a^2 b^2 c^2+b^4 c^2-a^2 c^4+b^2 c^4-c^6 : : , SEARCH = 13.4518908001886, now X(46264) in ETC.