IMO 1977 LL CZS9

Let ABCD be a regular tetrahedron and Z an isometry map-

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IMO 1977 LL CZS9

Origin: CZS

Problem

Let ABCD be a regular tetrahedron and Z an isometry map- ping A, B, C, D into B, C, D, A, respectively. Find the set M of all points X of the face ABC whose distance from Z(X) is equal to a given number t. Find necessary and sufficient conditions for the set M to be nonempty.

Solution

A necessary and sufficient condition for M to be nonempty is that 1/ \sqrt 10 \leqt \leq1.