  When considering geometry, one might think of working with lines and circles
on a flat plane as in Euclidean geometry. However, doing geometry in other
spaces is possible, as the existence of spherical and hyperbolic geometry
demonstrates. Despite the differences between these three geometries, striking
connections appear among the three. In this paper, we illuminate one such
connection by generalizing the cross-ratio, a powerful invariant associating a
number to four points on a line, into non-Euclidean geometry. Along the way, we
see how projections between these geometries can allow us to directly export
results from one geometry into the others. The paper culminates by generalizing
Carnot's Theorem for Conics - a classical result relating when six points on a
triangle lie on a conic - into spherical and hyperbolic geometry. These same
techniques are then applied to Carnot's Theorem for higher degree curves.
