Which proofs are you referring to?
Discussion
I was thinking of Girard’s Theorem, but there are others that deal with the geometry of spherical triangles. And then there’s Eratosthenes but that’s slightly more oblique.
So how would you prove the earth is a sphere from Girard’s Theorem?
Eratosthenes works to argue for a sphere if you assume the earth is a sphere and the sun is millions of miles away. But it could also be used to prove a flat earth if you assume the sun is small and more local and the earth is flat.
It's a trivial proof to dissambiguate planar and spherical triangles and has been perfomed many times since Girard's time. The theorem (A=R2(α+β+γ−π)) requires that the sum of the internal angles of a spherical triangle must exceed 180 degrees if that triangle is to have a positive area.
So... Draw a triangle (ABC) on the surface of the earth. Let the triangle be as large as possible, with each side several hundred km long. Measure the internal angles of the triangle you have constructed and find the sum. Is it greater than 180 degrees? If so, you have demonstrated that the triangle is spherical. If not, you have proven that the triangle is planar.
It's worth pointing out that this procedure can only prove that the surface of the earth is spherical but cannot distinguish between the earth either being a complete sphere or an incomplete arc/portion of a sphere.