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K834

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X(2), X(3), X(30), X(110), X(5373), X(10620)

vertices of the Thomson triangle

excenters of the Thomson triangle

K834 is the Neuberg cubic of the Thomson triangle T. See the related K078, K764, K765.

It follows that K834 passes through the counterparts of all the points of K001 in T and obviously it has all the properties of K001 with respect to T. It is a circular cubic with focus X(74). The real asymptote is parallel at X(110) to the Euler line.

In particular, it is the locus of M such that the line passing through M and the centroid M' of the antipedal triangle of the isogonal conjugate (wrt ABC) of M is parallel to the Euler line.

K001 and K834 share the same points at infinity and meet again at six finite points (one of them is O) which lie on the rectangular hyperbola (H) with center G, passing through X(3), X(6), X(381), X(599), X(2574), X(2575) and the foci of the Steiner inellipse. (H) is the polar conic of O with respect to K005.

K834 is a member of the pencil generated by two decomposed cubics : one is the union of the Euler line and the circumcircle, the other is the union of the line at infinity and the Thomson-Jerabek hyperbola. This pencil also contains K463, K913.