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


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