Proposition #20 reads "In a circle the angle at the centre is double of the angle at the circumference, when the angles have the same circumference as base".
It seems to me, that if one considers (and it makes sense for one to consider) the diameter to be the straight angle "on the circumference" of the semicircle (either), that proposition #31's component, "the angle in the semicircle is right" immediately falls out, since the straight angle (diameter) would be double the right angle in the same semicircle.
Euclid's #31 actually does not even reference #20.
Am I wrong here, or could Euclid have proven that part of #31 directly from #20 in this way?
Euclid’s Elements book III proposition #31 direct from #20
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Best Answer
For Euclid, an "angle" is formed by two rays which are not part of the same line (see Book I Definition 8). So, to Euclid, a "straight angle" is not an angle at all, and so Proposition 31 is not a special case of Proposition 20 since Proposition 20 only applies when you have an angle at the center.