# Edge of the triangle

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Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle

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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.

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