It is well known that every construction that can be performed with compass and straightedge alone can also be performed using origami, see:
R. Geretschlager. Euclidean Constructions and the Geometry of Origami. Mathematics Magazine 68 (1995), no. 5, 357–371.
If one checks the paper by Geretschlager above, one sees that the construction for intersecting two circles is quite complex. Is there an easier construction?
The axioms for origami can be seen at:
Background: I teach a course in geometry for future teachers. With this construction (and other easy constructions), it would be fairly clear that every construction that can be performed with compass and straightedge alone can also be performed using origami.
I only have found two origami constructions for intersecting two circles - one in the above reference and another in a forgotten (!!!) reference. It was a strange book written in the 1950s about various geometric constructions.