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