To construct a square equal to a given rectilineal figure.
作一个正方形，使它等于给定直线形。
把给定直线形 A 化作矩形 BDE…（C 省略），其底 BE、高 ED；将 BE 延长到 F 使 EF=ED；以 BF 为直径的半圆与 DE 延长线交于 H。则 EH 上的正方形等于图形 A。
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 A be the given rectilineal figure; thus it is required to construct a square equal to the rectilineal figure A. For let there be constructed the rectangular parallelogram BD equal to the rectilineal figure A. [I. 45] Then, if BE is equal to ED, that which was enjoined will have been done; for a square BD has been constructed equal to the rectilineal figure A. But, if not, one of the straight lines BE, ED is greater.
先用 euclid-elements/book1-prop-045 把给定直线形化为等面积的矩形。
Let BE be greater, and let it be produced to F; let EF be made equal to ED, and let BF be bisected at G. With centre G and distance one of the straight lines GB, GF let the semicircle BHF be described; let DE be produced to H, and let GH be joined. Then, since the straight line BF has been cut into equal segments at G, and into unequal segments at E, the rectangle contained by BE, EF together with the square on EG is equal to the square on GF.
若矩形两边相等，它本身就是正方形。
[II. 5] But GF is equal to GH; therefore the rectangle BE, EF together with the square on GE is equal to the square on GH. But the squares on HE, EG are equal to the square on GH; [I. 47] therefore the rectangle BE, EF together with the square on GE is equal to the squares on HE, EG. Let the square on GE be subtracted from each; therefore the rectangle contained by BE, EF which remains is equal to the square on EH.
若两边不等，用第二卷关于矩形与平方的分割关系构造一条线，使其平方等于该矩形。
But the rectangle BE, EF is BD, for EF is equal to ED; therefore the parallelogram BD is equal to the square on HE. And BD is equal to the rectilineal figure A. Therefore the rectilineal figure A is also equal to the square which can be described on EH.
在这条线上作正方形（euclid-elements/book1-prop-046），即得等于给定直线形的正方形。