To construct one and the same figure similar to a given rectilineal figure and equal to another given rectilineal figure.
作一个图形，既与已知直线形相似，又与另一已知直线形等积。
已知三角形 ABC 与图形 D：在 BC 上贴 BE 等于 △ABC，在 CE 上贴 CM 等于 D；以 GH 为比例中项在 GH 上构造与 ABC 相似且面积等于 D 的图形 GHK。
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Let ABC be the given rectilineal figure to which the figure to be constructed must be similar, and D that to which it must be equal; thus it is required to construct one and the same figure similar to ABC and equal to D. Let there be applied to BC the parallelogram BE equal to the triangle ABC [I. 44], and to CE the parallelogram CM equal to D in the angle FCE which is equal to the angle CBL. [I. 45] Therefore BC is in a straight line with CF, and LE with EM.
在BC上作平行四边形BE等于三角形ABC，并在CE上作平行四边形CM等于D，使角FCE等于角CBL。
Now let GH be taken a mean proportional to BC, CF [VI. 13], and on GH let KGH be described similar and similarly situated to ABC. [VI. 18] Then, since, as BC is to GH, so is GH to CF, and, if three straight lines be proportional, as the first is to the third, so is the figure on the first to the similar and similarly situated figure described on the second, [VI. 19, Por.] therefore, as BC is to CF, so is the triangle ABC to the triangle KGH.
取BC与CF的比例中项GH，并在GH上作图形KGH与ABC相似且同向。
But, as BC is to CF, so also is the parallelogram BE to the parallelogram EF. [VI. 1] Therefore also, as the triangle ABC is to the triangle KGH, so is the parallelogram BE to the parallelogram EF; therefore, alternately, as the triangle ABC is to the parallelogram BE, so is the triangle KGH to the parallelogram EF. [V. 16] But the triangle ABC is equal to the parallelogram BE; therefore the triangle KGH is also equal to the parallelogram EF.
由比例性质，BC比CF等于三角形ABC比三角形KGH，也等于平行四边形BE比平行四边形EF。
But the parallelogram EF is equal to D; therefore KGH is also equal to D. And KGH is also similar to ABC.
因此三角形ABC与平行四边形BE相等，故三角形KGH与平行四边形EF相等，而EF等于D，所以KGH等于D且相似于ABC。