1973 USAMO Problems/Problem 1
Two points and lie in the interior of a regular tetrahedron . Prove that angle .
Let the side length of the regular tetrahedron be . Link and extend to meet the plane containing triangle at ; link and extend it to meet the same plane at . We know that and are inside triangle and that
Now let’s look at the plane containing triangle with points and inside the triangle. Link and extend on both sides to meet the sides of the triangle at and , on and on . We have
But since and are interior of the tetrahedron, points and cannot be both at the vertices and , . Therefore, .
Solution with graphs posted at
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
hurdler: Remark on solution 1: This proof is not rigorous, in the very last step. The last step needs more justification.
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