[1] Fix a triangle ABC and a

[2] Extend the cevians through X to intersect the circumcircle of ABC at points A*, B*, C* and define triangle A*B*C*.

[3] Apply repeatedly step [2] to produce a sequence of triangles T

The first picture below shows the corresponding sequence of triangles (first 11 terms) resulting from ABC and setting the triangle center equal to the

Here the sequence of triangles seems to have two accumulation points represented by two antipodal equilateral triangles inscribed in the circumcircle of ABC and having their sides parallel to the Napoleon triangle (green).

Here again the sequence of triangles seems to have two accumulation points represented by two antipodal equilateral triangles inscribed in the circumcircle of ABC and having their sides parallel to the Napoleon triangle (red).

The other well known triangle center is the circumcenter. This leads though to a trivial sequence of terms which are all equal alternatively to the triangle ABC itself and its antipodal.

A similar behaviour of the sequence is obtained for the symmedian point. The corresponding sequence alternates between the triangle ABC itself and A*B*C*, since the two triangles share the same symmedian point (see BrocardEllipse.html ).

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