You can prove the earth is a sphere by the geometry within it. Although the flat geometry is a good approximation to a certain extent (when distances are not great), when distances get large it fails. For example, on large scales in the earth, the sum of the angles of a triangle will sum more than 180 degrees. You need spheric geometry to describe geometrical properties of the earth that go beyond, say, 50-100 miles.

On the other hand, if the earth were flat, then all current notion of mechanics would fail, you would have to describe the world by some other set of theories that would come out quite more complicated. For example there would be almost no symmetry in the dynamics of solid bodies. Also, if the earth is flat, then the fact begs the question, are other planets flat? If they are, why do they always look round? For example, we could ask, how the hell is momentum conserved in the system of saturn and its disks?

A theory of a flat earth is not only inconvenient in describing the world, but i think it has several theoretical voids in it.