1978 USAMO Problems/Problem 4
(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular.
(b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?
(a) Let be the said tetrahedron, and let the inscribed sphere of touch the faces at . Then, are normals to the respective faces. We know that the angle between any two normals is equal, so we have at equal angles. Now, since and similar for the other sides, we have that is a regular tetrahedron. Now, the faces of are the tangent planes at , , , and . Then, consider a rotation about . The rotation sends , , and . Thus we have , , , and . Performing rotations about the other axes yields that has equal edges, so it is regular.
(b) Consider the normals . We can perform a transformation in which we slightly shift , , and closer such that is still equilateral, and such that all angles between every pair of normals is less that . Then, shift such that . Then five of the distances are equal but the sixth is not.
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