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X(5), X(3134)

infinite points of K024

complements of the vertices of the circumtangential triangle

K722 is the reciprocal polar transform of the Steiner deltoid H3 with respect to the nine point circle NPC. In other words, K722 is the locus of the poles in NPC of the Simson lines. It follows that K722 is tritangent to NPC.

K722 is an acnodal cubic with node X(5). It has three real inflexional asymptotes parallel to the sidelines of the Morley triangle. These are the images of the sidelines of the circumtangential triangle under the homothety with center X(3628), ratio -1/3.

K722 is invariant under the rotations with center X(5) and angles ±2π/3 and also under the symmetries in the axes of the deltoid.


Let M be a point on the circumcircle (O).

Denote by T(M) the tangent at M to (O), S(M) the Simson line of M, O(M) the orthopole of T(M) in ABC.

It is known that S(M) envelopes H3 and the contact of S(M) with H3 is O(M). Let P be the pole of S(M) in NPC, this point obviously on K722.

The tangent T(P) at P to K722 is then the polar of O(M) in NPC.

The parallel at X(5) to T(M) meets T(P) at P' which is the tangential of P in K722. The polar of P' in NPC contains O(M). It is the Simson line of the reflection of M about the parallel at X(3) to S(M).