elem.6.18
在给定线段上作一个与给定多边形相似且位置相似的多边形。
已知直线 AB 与直线形 CDEF：在 AB 上构造与 CDEF 相似且同向的图形 ABGH。
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 and CE the given rectilineal figure; thus it is required to describe on the straight line AB a rectilineal figure similar and similarly situated to the rectilineal figure CE. Let DF be joined, and on the straight line AB, and at the points A, B on it, let the angle GAB be constructed equal to the angle at C, and the angle ABG equal to the angle CDF. [I. 23] Therefore the remaining angle CFD is equal to the angle AGB; [I. 32] therefore the triangle FCD is equiangular with the triangle GAB.
连接DF，在线段AB上作角GAB等于角C，角ABG等于角CDF，则三角形FCD与GAB等角，对应边成比例。
Therefore, proportionally, as FD is to GB, so is FC to GA, and CD to AB. Again, on the straight line BG, and at the points B, G on it, let the angle BGH be constructed equal to the angle DFE, and the angle GBH equal to the angle FDE. [I. 23] Therefore the remaining angle at E is equal to the remaining angle at H; [I. 32] therefore the triangle FDE is equiangular with the triangle GBH; therefore, proportionally, as FD is to GB, so is FE to GH, and ED to HB.
在线段BG上作角BGH等于角DFE，角GBH等于角FDE，则三角形FDE与GBH等角，对应边成比例。
[VI. 4] But it was also proved that, as FD is to GB, so is FC to GA, and CD to AB; therefore also, as FC is to AG, so is CD to AB, and FE to GH, and further ED to HB. And, since the angle CFD is equal to the angle AGB, and the angle DFE to the angle BGH, therefore the whole angle CFE is equal to the whole angle AGH. For the same reason the angle CDE is also equal to the angle ABH.
由比例传递得FC:GA = CD:AB = FE:GH = ED:HB。
And the angle at C is also equal to the angle at A, and the angle at E to the angle at H. Therefore AH is equiangular with CE; and they have the sides about their equal angles proportional; therefore the rectilineal figure AH is similar to the rectilineal figure CE.
角CFE等于角AGH，角CDE等于角ABH，且角C等于角A，角E等于角H，故多边形AH与CE相似。