elem.6.5
如果两个三角形的边成比例，那么这两个三角形等角，且对应边所对的角相等。
三角形 ABC 与 DEF 三边对应成比例；在 EF 上点 E、F 处构造 ∠FEG=∠ABC、∠EFG=∠ACB 得辅助点 G，可证 G 与 D 重合，故 △ABC 与 △DEF 等角相似。
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Let ABC, DEF be two triangles having their sides proportional, so that, as AB is to BC, so is DE to EF, as BC is to CA, so is EF to FD, and further, as BA is to AC, so is ED to DF; I say that the triangle ABC is equiangular with the triangle DEF, and they will have those angles equal which the corresponding sides subtend, namely the angle ABC to the angle DEF, the angle BCA to the angle EFD, and further the angle BAC to the angle EDF. For on the straight line EF, and at the points E, F on it, let there be constructed the angle FEG equal to the angle ABC, and the angle EFG equal to the angle ACB; [I. 23] therefore the remaining angle at A is equal to the remaining angle at G. [I. 32] Therefore the triangle ABC is equiangular with the triangle GEF.
在直线EF上点E、F处作角FEG等于角ABC，角EFG等于角ACB，则三角形ABC与三角形GEF等角。
Therefore in the triangles ABC, GEF the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles; [VI. 4] therefore, as AB is to BC, so is GE to EF. But, as AB is to BC, so by hypothesis is DE to EF; therefore, as DE is to EF, so is GE to EF. [V. 11] Therefore each of the straight lines DE, GE has the same ratio to EF; therefore DE is equal to GE.
由等角三角形对应边成比例，得AB比BC等于GE比EF；又已知AB比BC等于DE比EF，故DE比EF等于GE比EF，因此DE等于GE。
[V. 9] For the same reason DF is also equal to GF. Since then DE is equal to EG, and EF is common, the two sides DE, EF are equal to the two sides GE, EF; and the base DF is equal to the base FG; therefore the angle DEF is equal to the angle GEF, [I. 8] and the triangle DEF is equal to the triangle GEF, and the remaining angles are equal to the remaining angles, namely those which the equal sides subtend. [I. 4] Therefore the angle DFE is also equal to the angle GFE, and the angle EDF to the angle EGF.
同理，DF等于GF。由边边边定理，三角形DEF全等于三角形GEF，故角DEF等于角GEF，角DFE等于角GFE，角EDF等于角EGF。
And, since the angle FED is equal to the angle GEF, while the angle GEF is equal to the angle ABC, therefore the angle ABC is also equal to the angle DEF. For the same reason the angle ACB is also equal to the angle DFE, and further, the angle at A to the angle at D; therefore the triangle ABC is equiangular with the triangle DEF.
因角GEF等于角ABC，故角ABC等于角DEF；同理角ACB等于角DFE，角BAC等于角EDF，所以三角形ABC与三角形DEF等角。