You can't really "prove" the shape of the Earth unless you were to acually go into space and see for yourself (Tom Bishop excluding, as obviously Round Earths look like Flat Earths).
The truth is that RE makes a whole lot more sense as:
1) We don't have any crazy refraction.
2) Besides the fact that it supports life, the Earth is not special.
3) We don't have to alter the laws of perspective.
4) We don't have to include a conspiracy.
5) No Shadow Objects.
6) No confusing sun/moon relationships.
7) There is no order, order being an organization to the Universe, which is not natural. Things develop into chaos naturally.
8)No explanation needed for Great Circle Routes/travel times.
9) No "celestial gears" needed.
etc
PROBLEMS WITH RE:
1) We don't yet have an explanation for how gravity works.
Basically just gravity.
TOM BISHOP, don't give that stupid list, because 90% of the stuff on there deals with RE and FE, such as WIMPS, MACHOS, dark matter, etc.