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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.,–Lehmus_theorem.

EDIT: A large number of proofs, and a much larger number of references to proofs in the literature, can be found at

MORE EDIT: There are even proofs on YouTube: and

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