On a given straight line to describe a square.
在给定直线上作一个正方形。
在给定直线 AB 上作正方形 ABDE：先在 A 作 AB 的垂线 AC，截 AD=AB，过 D 作 DE 平行 AB，过 B 作 BE 平行 AD，构成正方形 ABDE（这里把 E 置于 B 的正上方）。
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 straight line; thus it is required to describe a square on the straight line AB. Let AC be drawn at right angles to the straight line AB from the point A on it [I. 11], and let AD be made equal to AB; through the point D let DE be drawn parallel to AB, and through the point B let BE be drawn parallel to AD. [I. 31] Therefore ADEB is a parallelogram; therefore AB is equal to DE, and AD to BE.
在给定直线 AB 的端点作垂线（euclid-elements/book1-prop-011）。
[I. 34] But AB is equal to AD; therefore the four straight lines BA, AD, DE, EB are equal to one another; therefore the parallelogram ADEB is equilateral. I say next that it is also right-angled. For, since the straight line AD falls upon the parallels AB, DE, the angles BAD, ADE are equal to two right angles.
在垂线上截取 AD 等于 AB，再过 D 作平行于 AB 的线，过 B 作平行于 AD 的线。
[I. 29] But the angle BAD is right; therefore the angle ADE is also right. And in parallelogrammic areas the opposite sides and angles are equal to one another; [I. 34] therefore each of the opposite angles ABE, BED is also right. Therefore ADEB is right-angled.
四边形 ABED 的对边平行且相等，角为直角。
And it was also proved equilateral. Therefore it is a square; and it is described on the straight line AB.
四边相等且四角为直角，所以它是给定直线上的正方形。