Rotation-a-Gons
Inspired by a Regolo Bizzi drawing (sooo much more elegant than this). This applet rotates a quadrilateral around a point and then draws shapes among the intersections. Pretty fun to play with; on GeoGebraTube.
The option checkbox does polytopes on all the intersections, and the more checkbox sometimes adds a few more.
There's a tool for making or adding your own. Select the center, then enter how many (N for the slider) and then three vertices for a triangle.