Let the edges of one of the faces of the tetrahedron have lengths , , and . Let , , and be the lengths of the sides that are not adjacent to the sides with lengths , , and , respectively.

Without loss of generality, assume that . I shall now prove that either or , by proving that if , then .

Assume that . The triangle inequality gives us that , so must be greater than . We also have from the triangle inequality that . Therefore . Therefore either or .

If , then the vertex where the sides of length , , and meet satisfies the given condition. If , then the vertex where the sides of length , , and meet satisfies the given condition. This proves the statement.