第3卷命题 21 · 同弧圆周角相等

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.

设圆ABCD，取圆心F，连接BF、FD。

For let the centre of the circle ABCD be taken, and let it be F; let BF, FD be joined.

角BFD为圆心角，角BAD为圆周角，它们以同一弧BCD为底，故角BFD是角BAD的两倍。

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.

同理，角BFD也是角BED的两倍。

[III. 20] For the same reason the angle BFD is also double of the angle BED; therefore the angle BAD is equal to the angle BED.

因此，角BAD等于角BED。