Six Triangle Orbits
Consider a triangle with sides labeled , , and a point and its trajectory as it visits each side of the triangle in some prespecified order, always taking the shortest route (by side we mean the infinite line resulting from extending the actual side of the triangle in both directions.)[more]
We may iterate this process and notice that the trajectory converges to another triangle similar to the initial one.
The two limit triangles formed from using a side order and its reverse appear congruent and their six vertices are concentric.
This Demonstration lets you drag the vertices of the triangle, the position of point , and the visiting order so that you can investigate this phenomenon.[less]
 M. Carvalho and M. Hager, "Geometric Orbits," The Mathematical Intelligencer, 34(2), 2012 pp. 56–62. doi:10.1007/s00283-012-9287-y.