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X(30), X(5627)

traces of X(30) on ABC sidelines

6 intersections of the cevian lines of X(13) with those of X(14)

The Tixier cubic is the only pK60+ with pivot the infinite point X(30) of the Euler line. See Special Isocubics ยง6.6.3. It is a member of the class CL006 of pK60+ cubics.

The asymptotes concur at X(476) = Tixier point. One of them is parallel to the Euler line and meets K037 again at X(5627). The satellite line of the line at infinity is the parallel at X(5627) to the Fermat line (light blue line on the figure below).

The pole of K037 is the point X(1989), the barycentric product of the Fermat point and the isogonal conjugate of X(323).

The polar conic of X(30) in both Tixier and Neuberg cubics is the rectangular diagonal hyperbola through the in/excenters and X(5). Its center is X(476).

The isogonal transform of K037 is K374. See also K205 and a generalization at K508.


The hessian cubic of K037 is K958, a remarkable focal cubic with singular focus X(476) passing through X(13), X(14), X(542).

The polar conic of any point P in K037 is a rectangular hyperbola that degenerates into two perpendicular lines when P lies on the hessian.

Those of X(13), X(14), X(542) belong to a same pencil passing through the poles of the Fermat line in K037 (green points). These poles are the in/excenters of the triangle X(13)X(14)X(476).

Naturally, K037 meets its hessian at their nine common inflexion points, three of them being real and collinear (blue points).

Compare the hessian cubics of K037 and K205.

See the page K958 for further properties.