If a straight line be cut at random, the square on the whole and that on one of the segments both together are equal to twice the rectangle contained by the whole and the said segment and the square on the remaining segment.
若一条直线任意分割，则整线和一段上的两个正方形，等于两倍整线与该段所成矩形加余段上的正方形。
正方形 ADEB 建于 AB，C 把 AB 分成两段。过 C 作 CF 平行于 AD。CB 上的小正方形（CGEB 内一角）含 G 顶点，DG 区域是 AC 上的正方形。gnomon KLM 围绕这两个正方形之外的部分。
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 let a straight line AB be cut at random at the point C; I say that the squares on AB, BC are equal to twice the rectangle contained by AB, BC and the square on CA. For let the square ADEB be described on AB, [I. 46] and let the figure be drawn.
把整线分成两段，在整线和目标段上分别作正方形。
Then, since AG is equal to GE, [I. 43] let CF be added to each; therefore the whole AF is equal to the whole CE. Therefore AF, CE are double of AF.
用分点把整线正方形分成两个矩形和两段平方。
But AF, CE are the gnomon KLM and the square CF; therefore the gnomon KLM and the square CF are double of AF. But twice the rectangle AB, BC is also double of AF; for BF is equal to BC; therefore the gnomon KLM and the square CF are equal to twice the rectangle AB, BC.
把同类矩形配对并使用 euclid-elements/book2-prop-003 的分解。
Let DG, which is the square on AC, be added to each; therefore the gnomon KLM and the squares BG, GD are equal to twice the rectangle contained by AB, BC and the square on AC. But the gnomon KLM and the squares BG, GD are the whole ADEB and CF, which are squares described on AB, BC; therefore the squares on AB, BC are equal to twice the rectangle contained by AB, BC together with the square on AC.
得到整线平方与目标段平方之和等于两倍整线与目标段矩形加余段平方。