Prove that in triangle $ABC$ if angle bisectors drawn from $B$ and $C$ are congruent then $AB=AC$.
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This is the Steiner-Lehmus Theorem. See, e.g., http://en.wikipedia.org/wiki/Steiner–Lehmus_theorem.
EDIT: A large number of proofs, and a much larger number of references to proofs in the literature, can be found at http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/steiner-lehmus.