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X(3), X(512), X(2574), X(2575) other points below 

Geometric properties : 

Each cubic of the Euler pencil has an osculating circle (Co) at O whose center lies on K920. The figure above shows the McCay cubic K003 with its corresponding (blue) circle passing through O, X(74), X(476) whose center is labelled Ω obviously on the perpendicular at O to the Euler line. Ω is the barycentric product of X(523) and the barycentric square of X(323). More about Ω below. K920 is an acnodal cubic with singularity at O. It has three real asymptotes : • two are parallel to the asymptotes of the Jerabek hyperbola and meet at X which lies on K920. X is the homothetic of X(6) under h(X3, – 1/4). • the third one is perpendicular to the Brocard axis at X', the reflection of X about O or the homothetic of X(6) under h(X3, 1/4).
The tangential Y of X lies on the third asymptote and on the line X(3)X(67). It follows that the line (L) passing through X, Y is the satellite line of the line at infinity. These points Ω, X, Y are now X(14809), X(14810), X(14811) in ETC (20171010). The cubics corresponding to X, Y have unlisted pivots with SEARCH numbers 96.4647947120816, 572.554539745838 respectively. X(512) is obtained with the Darboux cubic K004 since (Co) splits into the line at infinity and the Brocard axis. X(2574), X(2575) are obtained with the two cubics of the Euler pencil which have an inflexion point at O. See Table 27, section : Points on these cubics, (6). *** Consider the pencil of pKs generated by K003 and the union (U) of the cevian lines of O. All the cubics of the pencil – apart (U) – have the same tangent, polar conic, osculating circle at O which is precisely the circle with center Ω passing through O as above. This pencil also contains K073, K172, K349, pK(X571, X3) = K690*, pK(X3, X3).
