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PonceletPorism

The gray triangle *ABC* has circumcircle and incircle as shown. As the point *t* goes around the incircle, the tan-colored triangle "rotates" through the set of **all** triangles that have this same circumcircle and incircle, thus solving a famous problem called the "Poncelet porism". During this "rotation", points such as the orthocenter, labeled *H*, produce orbits. Shown here are the orbits of *H* and of the midpoint of a side of the tan-colored triangle.
This configuration (and others) show that *H* (and other special points) are really functions, not merely points. For example, the domain of *H* is the set of all triangles.

For more, see *GEOMETRY IN ACTION*, Chapter 7.