AoPSWiki
Want to learn how to tackle those tough AMC/AIME/Olympiad algebra problems? Check out Art of Problem Solving's Intermediate Algebra by Richard Rusczyk and Mathew Crawford. Over 1600 problems!

Pythagorean Inequality

From AoPSWiki

The Pythagorean Inequality is a generalization of the Pythagorean Theorem. The Theorem states that in a right triangle with sides of length a \leq b \leq c we have a^2 + b^2 = c^2. The Inequality extends this to obtuse and acute triangles. The inequality says:

For an acute triangle with sides of length a \leq b \leq c, a^2+b^2>c^2. For an obtuse triangle with sides a \leq b \leq c, a^2+b^2<c^2.

This inequality is a direct result of the Law of Cosines, although it is also possible to prove without using trigonometry.

See also

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/Pythagorean_Inequality"
Trying to get to the USAMO in 2010? Our AIME Problem Series can help you get there! Click here to enroll today!
© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us