第6卷命题 14 · 等角平行四边形边成反比则等积

Let AB, BC be equal and equiangular parallelograms having the angles at B equal, and let DB, BE be placed in a straight line; therefore FB, BG are also in a straight line. [I. 14] I say that, in AB, BC, the sides about the equal angles are reciprocally proportional, that is to say, that, as DB is to BE, so is GB to BF. For let the parallelogram FE be completed.

设AB、BC是相等且等角的平行四边形，角B相等，且DB、BE在一直线上，则FB、BG也在一直线上。

Since, then, the parallelogram AB is equal to the parallelogram BC, and FE is another area, therefore, as AB is to FE, so is BC to FE. [V. 7] But, as AB is to FE, so is DB to BE, [VI. 1] and, as BC is to FE, so is GB to BF.

断言在AB、BC中，等角的两边成反比，即DB比BE等于GB比BF。为此，补全平行四边形FE。

[id.] therefore also, as DB is to BE, so is GB to BF. [V. 11] Therefore in the parallelograms AB, BC the sides about the equal angles are reciprocally proportional.

由于AB等于BC，且FE是另一图形，故AB比FE等于BC比FE。但AB比FE等于DB比BE，BC比FE等于GB比BF，因此DB比BE等于GB比BF。

Next, let GB be to BF as DB to BE; I say that the parallelogram AB is equal to the parallelogram BC. For since, as DB is to BE, so is GB to BF, while, as DB is to BE, so is the parallelogram AB to the parallelogram FE, [VI. 1] and, as GB is to BF, so is the parallelogram BC to the parallelogram FE, [VI. 1] therefore also, as AB is to FE, so is BC to FE; [V. 11] therefore the parallelogram AB is equal to the parallelogram BC.

反之，设GB比BF等于DB比BE，则AB比FE等于BC比FE，故AB等于BC。