and a number k that represents the ratio of the homothety, a homothety is a transformation that will move P to a point P' where P' is on the ray AP and dE(A,P') = k·dE(A,P). Three dunce cap theorem Consider the three circles ω1, ω2, and ω3 in a common plane. Let P1 be the intersection of the two external tangent lines of ω2 and ω3. Let P2 and P3 be found the same way. The Three Dunce Caps Theorem then says that P1, P2, and P3 all lie on the same line.Proof:'' Construct a sphere on top of each circle and then construct a plane tangent to these three spheres. The plane intersects the plane that the circles lies on at a straight line containing P1, P2, and P3. These points are also the centers of homothety for the circles that they were derived from. Applying to spherical geometry Algebraically, hyperbolic and spherical geometry have the same structure. This allows us to apply concepts and theorems to one geometry to the other. Applying hyperbolic geometry to spherical geometry can make it easier to understand because spheres are much more concrete, which then makes spherical geometry easier to conceptualize. References Hyperbolic geometry