If a straight line be cut at random, the rectangle contained by the whole and one of the segments is equal to the rectangle contained by the segments and the square on the aforesaid segment.
若一条直线任意分割，则整线与其中一段所成矩形等于两段所成矩形加该段上的正方形。
C 把 AB 分成两段；在 CB 上作正方形 CDEB，将 ED 延长到 F，AF 平行于 CD 或 BE，整线与一段所成矩形 AE 被切成 AD 与 CE 两块。
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For let the straight line AB be cut at random at C; I say that the rectangle contained by AB, BC is equal to the rectangle contained by AC, CB together with the square on BC. For let the square CDEB be described on CB; [I. 46] let ED be drawn through to F, and through A let AF be drawn parallel to either CD or BE.
设整线分成两段，其中一段为目标段。
[I. 31] Then AE is equal to AD, CE.
整线与目标段所成矩形，可按另一方向的分点分成两个部分。
Now AE is the rectangle contained by AB, BC, for it is contained by AB, BE, and BE is equal to BC; AD is the rectangle AC, CB, for DC is equal to CB; and DB is the square on CB.
一个部分是两段所成矩形，另一个部分是目标段上的正方形。
Therefore the rectangle contained by AB, BC is equal to the rectangle contained by AC, CB together with the square on BC.
所以整线与该段所成矩形等于两段矩形加该段平方。