SequenceOfTriangles

Triangle Sequences

The next pictures show the first few terms of sequences of triangles build by applying to a basic triangle ABC the following recipe:
[1] Fix a triangle ABC and a triangle center X (f.e the centroid or the orthocenter etc.).
[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 T₀, ..., T_n, ... where each T_i+1 results from T_i in the same way A*B*C* results from ABC.

The first picture below shows the corresponding sequence of triangles (first 11 terms) resulting from ABC and setting the triangle center equal to the orthocenter. It seems that the sequence of triangles is divergent.

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 ).

Triangle Sequences

See Also