Prove that the tetrahedron with vertices at the points O(0, 0, 0), A(0, 1, 1), B(1, 0, 1) and C(1, 1, 0) is a regular one.

A regular tetrahedron is that in which the faces are equilateral triangles .

Take triangle OAB 
So OA = $\sqrt{(0-0{)}^2 +(0-1{)}^2+(0-1{)}^2}=\sqrt{2}$
OB = $\sqrt{2}$
AB = $\sqrt{2}$
Hence this face is equilateral .

Similarly take other faces, you will get all faces as equilateral triangles .
Hence the tetrahedron is regular