Circular continuity principle:
If a circle C has one point inside and one point outside another circle C' , then the two circles intersect in two distinct points.
I read this on Euclidean and non-Euclidean geometries(by Greenberg) and I wanna prove it by Dedekind continuity axiom. I also have the segment-circle principle and Archimedes's axiom proven by Dedekind axiom.
Can someone please prove the circle continuity principle using what I have?
(Segment–circle continuity principle: If one endpoint of a segment is inside a circle and the other endpoint is outside, then the segment intersects the circle at a point in between.
ARCHIMEDES' AXIOM : If CD is any segment, A any point, and r any ray with vertex A, then for every point B on r, there is a number n such that when CD is laid off n times on r starting at A, a point E is reached such that n · CD is congruent to AE and either B = E or B is between A and E.)