To cut a given finite straight line in extreme and mean ratio.
将给定线段分为中外比，即较长部分与整条线段之比等于较短部分与较长部分之比。
将已知线 AB 按中外比（黄金分割）切分：以 AB 为边作正方形 ABDF，再贴一相等面积、亏形为正方形的平行四边形，得分点 E 使 AB:AE=AE:EB。
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 AB be the given finite straight line; thus it is required to cut AB in extreme and mean ratio. On AB let the square BC be described; and let there be applied to AC the parallelogram CD equal to BC and exceeding by the figure AD similar to BC.
设AB为给定线段，在AB上作正方形BC。
[VI. 29] Now BC is a square; therefore AD is also a square. And, since BC is equal to CD, let CE be subtracted from each; therefore the remainder BF is equal to the remainder AD.
在AC上作平行四边形CD，使其等于正方形BC，且超出部分AD与BC相似，故AD也是正方形。
But it is also equiangular with it; therefore in BF, AD the sides about the equal angles are reciprocally proportional; [VI. 14] therefore, as FE is to ED, so is AE to EB. But FE is equal to AB, and ED to AE.
因BC等于CD，减去公共部分CE，得余量BF等于AD，且等角，故对应边成比例：FE比ED等于AE比EB。
Therefore, as BA is to AE, so is AE to EB. And AB is greater than AE; therefore AE is also greater than EB.
因FE等于AB，ED等于AE，故BA比AE等于AE比EB，且AB大于AE，故AE大于EB。