conditions of properties

central and circumferential angle

A circumferential angle on a circular arc equals half of the central arc on the same circular arc. There's nothing as nice to illustrate this than an applet...

But is is always the case?

Sir, in my applet it isn't the case! Yes, if you drag C all the way down somewhere between A and B the property doesn't seem to be correct...

True or not?

Je could prevent this annoying situation by not defining C as a point on the circle. C= Point(CircularArc(M, B, A) ) defines C as a point on a circular arc. But by doing so you miss some opportunities. Defining C on a circle focusses on the importance of every word in a definition, here 'on the same circular arc'. If on the first view the property seemed to be wrong, it turns out to be an interesting case, and after exploring and calculating the correct angles the property still stands. So, what should you do? I'd say it depends on the class you're working in. What's interesting in one class in unnecessarily confusing in another. Never use GG in an automatic mode. There's no exclusive way to teach, neither with GG. But remind that working with GG you can differentiate and show things adapted to any class.