Porism of Poncelet
we have a closed polygon which is incribed in one conic, and has all its sides tangent to another. What is amazing is that if we arrange for this happen, there is an infinity of polygons satisfyng these two conditions with respect the same two conics. Illustrated just for a triangle.
Drage the vertex H around and see how the family of triangles share the same fixed in circle and ex-circles. Theres no symmetry operating here, the sides and angles all change.