第3卷命题 16 · 圆直径端垂线性质

Let ABC be a circle about D as centre and AB as diameter; I say that the straight line drawn from A at right angles to AB from its extremity will fall outside the circle. For suppose it does not, but, if possible, let it fall within as CA, and let DC be joined. Since DA is equal to DC, the angle DAC is also equal to the angle ACD. [I. 5] But the angle DAC is right; therefore the angle ACD is also right: thus, in the triangle ACD, the two angles DAC, ACD are equal to two right angles: which is impossible.

设圆ABC以D为圆心，AB为直径。从A作AB的垂线，假设它落在圆内如CA，连接DC。因DA等于DC，角DAC等于角ACD，但角DAC为直角，故角ACD也为直角，三角形ACD内角和等于两直角，矛盾。因此垂线不在圆内。

[I. 17] Therefore the straight line drawn from the point A at right angles to BA will not fall within the circle. Similarly we can prove that neither will it fall on the circumference; therefore it will fall outside. Let it fall as AE; I say next that into the space between the straight line AE and the circumference CHA another straight line cannot be interposed. For, if possible, let another straight line be so interposed, as FA, and let DG be drawn from the point D perpendicular to FA.

同理可证垂线也不在圆周上，故必落在圆外，如AE。

Then, since the angle AGD is right, and the angle DAG is less than a right angle, AD is greater than DG. [I. 19] But DA is equal to DH; therefore DH is greater than DG, the less than the greater: which is impossible. Therefore another straight line cannot be interposed into the space between the straight line and the circumference. I say further that the angle of the semicircle contained by the straight line BA and the circumference CHA is greater than any acute rectilineal angle, and the remaining angle contained by the circumference CHA and the straight line AE is less than any acute rectilineal angle.

假设在直线AE与圆弧CHA之间可插入另一条直线如FA，从D作DG垂直于FA。因角AGD为直角，角DAG小于直角，故AD大于DG。但DA等于DH，故DH大于DG，矛盾。因此无法插入另一条直线。

For, if there is any rectilineal angle greater than the angle contained by the straight line BA and the circumference CHA, and any rectilineal angle less than the angle contained by the circumference CHA and the straight line AE, then into the space between the circumference and the straight line AE a straight line will be interposed such as will make an angle contained by straight lines which is greater than the angle contained by the straight line BA and the circumference CHA, and another angle contained by straight lines which is less than the angle contained by the circumference CHA and the straight line AE. But such a straight line cannot be interposed; therefore there will not be any acute angle contained by straight lines which is greater than the angle contained by the straight line BA and the circumference CHA, nor yet any acute angle contained by straight lines which is less than the angle contained by the circumference CHA and the straight line AE. — PORISM.

若存在直线角大于BA与圆弧CHA所夹角，或小于圆弧CHA与AE所夹角，则可在圆弧与AE之间插入一直线，产生更大的直线角和更小的直线角，但已证无法插入，故半圆角大于任何锐直线角，剩余角小于任何锐直线角。