Question

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.

Answer 1

The faces of a regular tetrahedron are equilateral triangles.

Let us consider $△$OAB
$$\begin{gathered} OA=\sqrt{{\left(0-0\right)}^2+{\left(0-1\right)}^2+{\left(0-1\right)}^2} \\ =\sqrt{2} \end{gathered}$$
$$\begin{gathered} OB=\sqrt{{\left(1-0\right)}^2+{\left(0-0\right)}^2+{\left(1-0\right)}^2} \\ =\sqrt{2} \end{gathered}$$
$$\begin{gathered} AB=\sqrt{{\left(1-0\right)}^2+{\left(0-1\right)}^2+{\left(1-1\right)}^2} \\ =\sqrt{2} \end{gathered}$$
Hence, this face is an equilateral triangle.
Similarly, $△$OBC, $△$OAC, $△$ABC all are equilateral triangles.
Hence, the tetrahedron is a regular one.