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. 

Bézier Curves - How the Recursive Algorithm Works

A set of n+1 points will create a Bézier curve of n-th order.

• Drag the first slider to select the number of points. 
• Drag the point on the slider "Recursion Level" to 1 to see the first new set of points. 
• Continue dragging the Recursion Level slider to the end to see all recursive steps and how the last point, generating the Bézier curve, is constructed.
•  Hit the Play button to follow the generating point on the Bézier Curve. 

Irina Boyadzhiev, Created with GeoGebra

More about the construction and other Bezier curve constructions at: MyGeoGebra

Download Applet

Angle Bisector of a Right Triangle

Move point P.  What do you observe?

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

1. What can you say about the diagonal of the outer square in relation to its angles? in relation to the right triangles formed by the square? 
2. What can you say about the diagonal of the outer square in relation to the inner square?
3. What conjecture can you make from your observation in 2?

Trisecting a Segment

Move the point on the slider to the right to perform geometric construction.

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

Can you see why AG is one third of AB?