elem.3.27
在等圆中，立于等弧上的角相等，无论它们是圆心角还是圆周角。
本页以“等圆中等弧对等角”整体图解辅助阅读；点、线、角、圆索引已按命题文字和证明步骤校订，可与证明和问答联动。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
For in equal circles ABC, DEF, on equal circumferences BC, EF, let the angles BGC, EHF stand at the centres G, H, and the angles BAC, EDF at the circumferences; I say that the angle BGC is equal to the angle EHF, and the angle BAC is equal to the angle EDF. For, if the angle BGC is unequal to the angle EHF, one of them is greater.
设等圆ABC和DEF，等弧BC和EF，圆心角BGC和EHF，圆周角BAC和EDF。
Let the angle BGC be greater : and on the straight line BG, and at the point G on it, let the angle BGK be constructed equal to the angle EHF. [I. 23] Now equal angles stand on equal circumferences, when they are at the centres; [III. 26] therefore the circumference BK is equal to the circumference EF.
假设角BGC不等于角EHF，则一角较大。设角BGC较大，在直线BG上点G处作角BGK等于角EHF。
But EF is equal to BC; therefore BK is also equal to BC, the less to the greater : which is impossible. Therefore the angle BGC is not unequal to the angle EHF; therefore it is equal to it.
根据III.26，等圆心角对等弧，故弧BK等于弧EF。但EF等于BC，因此BK等于BC，小等于大，矛盾。
And the angle at A is half of the angle BGC, and the angle at D half of the angle EHF; [III. 20] therefore the angle at A is also equal to the angle at D.
所以角BGC等于角EHF。又根据III.20，角A是角BGC的一半，角D是角EHF的一半，故角A等于角D。