# 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

as desired.