The Arithmetic Mean Geometric Mean Inequality

Move A along the circumference of the semi-circle. What can you say about the relationship between AD and the radius of the circle?

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

The applet above shows the geometric proof of the arithmetic mean geometric mean inequality. The diameter of the circle is a + b and the radius is (a + b)/2. The length AD is less than or equal to the radius of the circle.  The complete explanation of the inequality can be found here.