第6卷命题 19 · 相似三角形面积比定理

Let ABC, DEF be similar triangles having the angle at B equal to the angle at E, and such that, as AB is to BC, so is DE to EF, so that BC corresponds to EF; [V. Def. 11] I say that the triangle ABC has to the triangle DEF a ratio duplicate of that which BC has to EF. For let a third proportional BG be taken to BC, EF, so that, as BC is to EF, so is EF to BG; [VI. 11] and let AG be joined. Since then, as AB is to BC, so is DE to EF, therefore, alternately, as AB is to DE, so is BC to EF.

设相似三角形ABC和DEF，角B等于角E，且AB:BC = DE:EF，则BC对应EF。

[V. 16] But, as BC is to EF, so is EF to BG; therefore also, as AB is to DE, so is EF to BG. [V. 11] Therefore in the triangles ABG, DEF the sides about the equal angles are reciprocally proportional. But those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal; [VI. 15] therefore the triangle ABG is equal to the triangle DEF.

取BC、EF的第三比例项BG，使BC:EF = EF:BG，连接AG。

Now since, as BC is to EF, so is EF to BG, and, if three straight lines be proportional, the first has to the third a ratio duplicate of that which it has to the second, [V. Def. 9] therefore BC has to BG a ratio duplicate of that which CB has to EF. But, as CB is to BG, so is the triangle ABC to the triangle ABG; [VI. 1] therefore the triangle ABC also has to the triangle ABG a ratio duplicate of that which BC has to EF. But the triangle ABG is equal to the triangle DEF; therefore the triangle ABC also has to the triangle DEF a ratio duplicate of that which BC has to EF.

由AB:BC = DE:EF，交替得AB:DE = BC:EF，又BC:EF = EF:BG，故AB:DE = EF:BG，因此三角形ABG与DEF等积。

Therefore etc. PORISM.

因BC:EF = EF:BG，故BC:BG为BC:EF的二次比，而三角形ABC与ABG面积比等于BC:BG，所以三角形ABC与DEF面积比为BC:EF的二次比。