1991 IMO Problems/Problem 1
Given a triangle let be the center of its inscribed circle. The internal bisectors of the angles meet the opposite sides in respectively. Prove that
We have . From Van Aubel's Theorem, we have which from the Angle Bisector Theorem reduces to . We find similar expressions for the other terms in the product so that the product simplifies to . Letting for positive reals , the product becomes . To prove the right side of the inequality, we simply apply AM-GM to the product to get
To prove the left side of the inequality, simply multiply out the product to get
Remark: To prove the right side of the inequality, a quicker way might be to use Gergonne's and AM-GM.
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