Book 3
25
Similarly, even if the angle ABD be equal to the angle BAD, AD being equal to each of the two BD, DC, the three straight lines DA, DB, DC will be equal to one another, D will be the centre of the completed circle, and ABC will clearly be a semicircle.
25
But, if the angle ABD be less than the angle BAD, and if we construct, on the straight line BA and at the point A on it, an angle equal to the angle ABD, the centre will fall on DB within the segment ABC, and the segment ABC will clearly be greater than a semicircle.
25
Therefore, given a segment of a circle, the complete circle has been described. Q. E. F. to describe the complete circle, προσαναγράψαι τὸν κύκλον, literally “to describe the circle on to it.’
PROPOSITION 26.
26
In equal circles equal angles stand on equal circumferences, whether they stand at the centres or at the circumferences.
26
Let ABC, DEF be equal circles, and in them let there be equal angles, namely at the centres the angles BGC, EHF, and at the circumferences the angles BAC, EDF; I say that the circumference BKC is equal to the circumference ELF.
26
For let BC, EF be joined.
26
Now, since the circles ABC, DEF are equal, the radii are equal.
26
Thus the two straight lines BG, GC are equal to the two straight lines EH, HF; and the angle at G is equal to the angle at H; therefore the base BC is equal to the base EF. [I. 4]
26
And, since the angle at A is equal to the angle at D, the segment BAC is similar to the segment EDF; [III. Def. 11] and they are upon equal straight lines.
26
But similar segments of circles on equal straight lines are equal to one another; [III. 24] therefore the segment BAC is equal to EDF. But the whole circle ABC is also equal to the whole circle DEF; therefore the circumference BKC which remains is equal to the circumference ELF.
26
Therefore etc. Q. E. D.
PROPOSITION 27.
27
In equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centres or at the circumferences.
