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K764

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X(2), X(3), X(6), X(376), X(1350), X(5373), X(9740), X(9741)

A', B', C' : excenters of the Thomson triangle T = Q1Q2Q3, the incenter being X(5373)

their reflections about X(3)

S1, S2, S3 : reflections of Q1, Q2, Q3 about X(3)

infinite points of the Thomson cubic K002

K764 is the Darboux cubic for the Thomson triangle T = Q1Q2Q3. Hence, it is a member of the Euler pencil of cubics in T also containing K078 = McCay cubic of T, K765 = Thomson cubic of T, K834 = Neuberg cubic of T and K615, the only cubic which is a circum-cubic in ABC.

K764 is a central cubic with center X(3) and asymptotes parallel at X(3) to those of the Thomson cubic K002. K764 and K002 meet at six finite points lying on the Jerabek hyperbola JT of the Thomson triangle namely X(2), X(3), X(6), Q1, Q2, Q3.

K764 passes through the reflections S1, S2, S3 of Q1, Q2, Q3 about X(3). These are the points on (O) whose Simson lines concur at X(2). These lines are the altitudes of the Thomson triangle. The tangents to K764 at S1, S2, S3 concur at X(1350).

K764 also passes through the reflections of X(5373), A', B', C' about X(3). These four points lie on the Stammler hyperbola and the two remaining common points are X(3), X(6).

The pivot and isopivot of K764 in T are X(376) and X(6) respectively, hence K764 passes through the vertices of the cevian triangle of X(376) in T and the tangents at X(376), Q1, Q2, Q3 concur at X(6).