To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment.
分割给定线段，使整线与其中一段所成矩形等于余段上的正方形。
在 AB 上作正方形 ABDC（A 左下、B 右下、D 右上、C 左上）。E 是 AC 中点；将 CA 延长至 F 使 EF=EB；在 AF 上作正方形 FGHA（G 在底）。G H 延长交 CD 于 K。H 即所求分点：AB·BH = AH²。
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 cut AB so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment. For let the square ABDC be described on AB; [I. 46] let AC be bisected at the point E, and let BE be joined; let CA be drawn through to F, and let EF be made equal to BE; let the square FH be described on AF, and let GH be drawn through to K. I say that AB has been cut at H so as to make the rectangle contained by AB, BH equal to the square on AH.
在给定线段上作正方形，并按 euclid-elements/book2-prop-006 的关系寻找分点。
For, since the straight line AC has been bisected at E, and FA is added to it, the rectangle contained by CF, FA together with the square on AE is equal to the square on EF. [II. 6] But EF is equal to EB; therefore the rectangle CF, FA together with the square on AE is equal to the square on EB.
让整线与一段所成矩形同余段平方相等。
But the squares on BA, AE are equal to the square on EB, for the angle at A is right; [I. 47] therefore the rectangle CF, FA together with the square on AE is equal to the squares on BA, AE. Let the square on AE be subtracted from each; therefore the rectangle CF, FA which remains is equal to the square on AB.
构造中通过半线和延长关系把矩形差转成正方形。
Now the rectangle CF, FA is FK, for AF is equal to FG; and the square on AB is AD; therefore FK is equal to AD. Let AK be subtracted from each; therefore FH which remains is equal to HD.
所得分点即满足整线矩形等于余段平方。