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
Solution
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.
Remark: To prove the right side of the inequality, a quicker way might be to use Gergonne's and AM-GM.
See Also
1991 IMO (Problems) • Resources | ||
Preceded by First Question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |