elem.3.31
在一个圆中，半圆内的角是直角；大于半圆的弧段内的角小于直角；小于半圆的弧段内的角大于直角。此外，大弧段的角大于直角，小弧段的角小于直角。
本页以“半圆内角为直角”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。
Let ABCD be a circle, let BC be its diameter, and E its centre, and let BA, AC, AD, DC be joined; I say that the angle BAC in the semicircle BAC is right, the angle ABC in the segment ABC greater than the semicircle is less than a right angle, and the angle ADC in the segment ADC less than the semicircle is greater than a right angle. Let AE be joined, and let BA be carried through to F. Then, since BE is equal to EA, the angle ABE is also equal to the angle BAE.
连接圆心E与A，并延长BA至F。由于BE等于EA，故角ABE等于角BAE；同理，CE等于EA，故角ACE等于角CAE。因此角BAC等于角ABC与角ACB之和。
[I. 5] Again, since CE is equal to EA, the angle ACE is also equal to the angle CAE. [I. 5] Therefore the whole angle BAC is equal to the two angles ABC, ACB. But the angle FAC exterior to the triangle ABC is also equal to the two angles ABC, ACB; [I. 32] therefore the angle BAC is also equal to the angle FAC; therefore each is right; [I. Def. 10] therefore the angle BAC in the semicircle BAC is right.
三角形ABC的外角FAC等于两内对角ABC与ACB之和，故角BAC等于角FAC，因此两者均为直角。所以半圆BAC内的角BAC是直角。
Next, since in the triangle ABC the two angles ABC, BAC are less than two right angles, [I. 17] and the angle BAC is a right angle, the angle ABC is less than a right angle; and it is the angle in the segment ABC greater than the semicircle. Next, since ABCD is a quadrilateral in a circle, and the opposite angles of quadrilaterals in circles are equal to two right angles, [III. 22] while the angle ABC is less than a right angle, therefore the angle ADC which remains is greater than a right angle; and it is the angle in the segment ADC less than the semicircle. I say further that the angle of the greater segment, namely that contained by the circumference ABC and the straight line AC, is greater than a right angle; and the angle of the less segment, namely that contained by the circumference ADC and the straight line AC, is less than a right angle.
在三角形ABC中，两角ABC与BAC之和小于两直角，而角BAC为直角，故角ABC小于直角，此即大于半圆的弧段ABC内的角。
This is at once manifest. For, since the angle contained by the straight lines BA, AC is right, the angle contained by the circumference ABC and the straight line AC is greater than a right angle. Again, since the angle contained by the straight lines AC, AF is right, the angle contained by the straight line CA and the circumference ADC is less than a right angle.
由于ABCD是圆内接四边形，对角互补，而角ABC小于直角，故对角ADC大于直角，此即小于半圆的弧段ADC内的角。另外，由直线BA、AC所夹的直角可知，弧ABC与弦AC所夹的角大于直角；由直线AC、AF所夹的直角可知，弦CA与弧ADC所夹的角小于直角。