Tetrahedron is minimum energy conformation for atom with 4 electron domains in VSEPR theory
Sketch of a Simpler Solution
There is one remaining way that is simpler than all of the above, and it has to do with circumscribing a tetrahedron with a cube.
I will leave it to the reader to work on this most beautiful version. The properties of the cube ensure that the distances are maximized, and some more geometry will show that the inscribed shape is indeed a regular tetrahedron. For a hint, check out this excellent answer to this question on the math stack exchange site.