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

K205 is an equilateral nK0+ with root X(523) and pole X(1989). See the related K501.

The three real asymptotes concur at X = X(14846), on the lines X(2)X(249) and X(111)X(265). See also K037 and a generalization at K508.

K205a

The hessian cubic of K205 is a remarkable focal cubic with singular focus X passing through X(13), X(14), X(115), X(476).

The polar conic of any point P in K205 is a rectangular hyperbola that degenerates into two perpendicular lines when P lies on the hessian.

Those of X(13), X(14), X(115) belong to a same pencil passing through the poles of the Fermat line in K205 (green points). These poles are the in/excenters of the triangle X(13)X(14)X(476).

Naturally, K205 meets its hessian at their nine common inflexion points, three of them being real and collinear (blue points).

K205b

Compare the hessian cubics of K037 (red curve) and K205 (blue curve).

These two focal cubics pass through X(476), X(13), X(14) and have the same tangents passing through X(476) at these two latter points.