It's well known that you can't trisect an arbitrary angle with a traditional straight-edge and compass. However, using methods in traditional Origami, we can trisect an angle. Typically, when discussing these two topics, it's usually in light of demonstrating things that can be done with Origami that can't be done with Euclidean Geometry. However, one notable fact is that you can't construct a circle via techniques in Origami.
So for example, the famous $9$-point circle theorem can't be achieved by methods of Origami. Are there other notable constructions in Euclidean Geometry that can't be accomplished in Origami Geometry?