Seventh circle theorem

First page
Theorem 1
Theorem 2
Theorem 3
Theorem 4
Theorem 5
Theorem 6
Theorem 7
The Seven Theorems

Alternate Segment Theorem

Seventh circle theorem:

(I find this one the oddest and most unexpected theorem of them all!!)
What's the connection between the two angles α and β?

You can use the cursor to move the point E around the circumference, but it doesn't change much, does it?
So you can try moving point F around the circumference, and hence alter the angles.
The connection between angles α and β should be clear!
Moving point C changes the position of the tangents.
Moving points A or B changes the size of the circle.

This is a Java Applet created using GeoGebra from - it looks like you don't have Java installed, please go to

If you need them, the seven circle theorems here

Tim Devereux, Created with GeoGebra