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.

Given: Points are O(0, 0, 0), A(0, 1, 1), B(1, 0, 1) and C(1, 1, 0)

To prove: given points are forming a regular tetrahedron

All edges of a regular tetrahedron are equal

Formula used:

The distance between any two points (a, b, c) and (m, n, o) is given by,

Therefore,

Distance between O(0, 0, 0) and A(0, 1, 1) is OA,

Distance between O(0, 0, 0) and B(1, 0, 1) is OB,

Distance between O(0, 0, 0) and C(1, 1, 0) is OC,

Distance between A(0, 1, 1) and B(1, 0, 1) is AB,

Distance between B(1, 0, 1) and C(1, 1, 0) is BC,

Distance between A(0, 1, 1) and C(1, 1, 0) is AC,

Clearly,

AB = BC = AC = OA = OB = OC

All edges are equal

Thus, A, B, C and O forms a regular tetrahedron

Hence Proved