Seventh circle theorem: 'Alternate Segment Theorem'

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

Now look at angles γ and δ in the window below - does the alternate segment theorem applies to them too?

If you need it, this theorem is set out here

and all the theorems are summarised here

Tim Devereux, Created with GeoGebra