elem.3.15
在圆中，直径是最大的线段，而其余弦中，越靠近圆心的弦总是大于更远的弦。
本页以“圆内直径最大弦近心者更大”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 AD be its diameter and E the centre; and let BC be nearer to the diameter AD, and FG more remote; I say that AD is greatest and BC greater than FG. For from the centre E let EH, EK be drawn perpendicular to BC, FG. Then, since BC is nearer to the centre and FG more remote, EK is greater than EH.
设圆ABCD，直径AD，圆心E，弦BC比FG更靠近直径AD。
[III. Def. 5] Let EL be made equal to EH, through L let LM be drawn at right angles to EK and carried through to N, and let ME, EN, FE, EG be joined. Then, since EH is equal to EL, BC is also equal to MN.
从E作EH、EK分别垂直于BC、FG，因BC更近圆心，故EK大于EH。
[III. 14] Again, since AE is equal to EM, and ED to EN, AD is equal to ME, EN. But ME, EN are greater than MN, [I. 20] and MN is equal to BC; therefore AD is greater than BC. And, since the two sides ME, EN are equal to the two sides FE, EG, and the angle MEN greater than the angle FEG, therefore the base MN is greater than the base FG.
作EL等于EH，过L作LM垂直于EK并延长至N，连接ME、EN、FE、EG。因EH等于EL，故BC等于MN。
[I. 24] But MN was proved equal to BC. Therefore the diameter AD is greatest and BC greater than FG.
因AE等于EM，ED等于EN，故AD等于ME加EN；而ME加EN大于MN，且MN等于BC，故AD大于BC。又因ME、EN等于FE、EG，且角MEN大于角FEG，故底MN大于底FG，而MN等于BC，所以直径AD最大，BC大于FG。