Book 3
20
Therefore the whole angle BEC is double of the whole angle BAC.
20
Again let another straight line be inflected, and let there be another angle BDC; let DE be joined and produced to G.
20
Similarly then we can prove that the angle GEC is double of the angle EDC, of which the angle GEB is double of the angle EDB; therefore the angle BEC which remains is double of the angle BDC.
20
Therefore etc. Q. E. D. let another straight line be inflected, κεκλάσθω δὴ πάλιν (without εὐθεῖα). The verb κλάω (to break off) was the regular technical term for drawing from a point a (broken) straight line which first meets another straight line or curve and is then bent back from it to another point, or (in other words) for drawing straight lines from two points meeting at a point on a curve or another straight line. κεκλάσθαι is one of the geometrical terms the definition of which must according to Aristotle be assumed (Anal. Post. I. 10, 76 b 9).
PROPOSITION 21.
21
In a circle the angles in the same segment are equal to one another.
21
Let ABCD be a circle, and let the angles BAD, BED be angles in the same segment BAED; I say that the angles BAD, BED are equal to one another.
21
For let the centre of the circle ABCD be taken, and let it be F; let BF, FD be joined.
21
Now, since the angle BFD is at the centre, and the angle BAD at the circumference, and they have the same circumference BCD as base, therefore the angle BFD is double of the angle BAD. [III. 20]
21
For the same reason the angle BFD is also double of the angle BED; therefore the angle BAD is equal to the angle BED.
21
Therefore etc. Q. E. D.
PROPOSITION 22.
22
The opposite angles of quadrilaterals in circles are equal to two right angles.
22
Let ABCD be a circle, and let ABCD be a quadrilateral in it; I say that the opposite angles are equal to two right angles.
22
Let AC, BD be joined.
22
Then, since in any triangle the three angles are equal to two right angles, [I. 32] the three angles CAB, ABC, BCA of the triangle ABC are equal to two right angles.
