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Translated task text
In this task you are going to work with certain type of proof for the (inscribed angel sentence?). The proof has three steps, as illustrated below. On the figures we have used (all those fancy greek letters). The inscribed angle sentence says $\alpha = \frac{\beta}{2}$. Write an explanation for the proof. Try to convey the idea behind each step, and after that the proof as a whole.

enter image description here

Now for the first circel I have the following.
$180-2\alpha$
$\beta = 180 - (180 -2\alpha)$
$\alpha = \frac{\beta}{2}$
This was fine, the proof here is spesific for when one chord is on the diameter. The scond picture you use what "discovered" in the first picture, and prove it for when S is exterior. On the third you don't use anything from the other steps.

Now I'm not sure what the question is here. I have gone through the individual proofs for each of these steps during other tasks in the book. I'm just a bit unsure about the proof as a whole. Is that just the point that with these three steps every possible layout of this figure has been covered? It's just so abstract. I'm reading on my own and would love any input.

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They are giving you the proof of three cases: the first two with $\,\alpha\,$ an acute angle and the last one with it an obtuse one. The first case is then subdivided in two cases: one of the angle's legs is a diameter, and then none of the legs is a diameter...I think this pretty much covers all the cases, doesn't it? –  DonAntonio Oct 17 '12 at 13:16
    
The German „Satz“ is usually translated as “theorem” in English, sometimes as “law” or “rule”. “Sentence” is rarely used in most areas of mathematics. –  MvG Oct 17 '12 at 14:12

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