If a point C lies between points A and B, can it be proven from Euclid's postulates that C lies on line AB?
This is not an area of mathematics that I'm particularly good at, but I'm pretty sure the answer is "no", unless you can define "lies between".
For example, you could argue that "lies between" means "has equal distance to", in which case the statement isn't even true, let alone you can prove it from Euclid's postulates.