Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.
等角平行四边形的面积比等于其两边比之复合比。
等角平行四边形 AC、CF 共顶点 C，BCG 共线、DCE 共线；下方比例链 K:L:M = BC:CG 把两平行四边形面积比化为对应边比的复合比。
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Let AC, CF be equiangular parallelograms having the angle BCD equal to the angle ECG; I say that the parallelogram AC has to the parallelogram CF the ratio compounded of the ratios of the sides. For let them be placed so that BC is in a straight line with CG; therefore DC is also in a straight line with CE. Let the parallelogram DG be completed; let a straight line K be set out, and let it be contrived that, as BC is to CG, so is K to L, and, as DC is to CE, so is L to M.
设等角平行四边形AC和CF，角BCD等于角ECG。将BC与CG置于同一直线上，则DC也与CE共线。
[VI. 12] Then the ratios of K to L and of L to M are the same as the ratios of the sides, namely of BC to CG and of DC to CE. But the ratio of K to M is compounded of the ratio of K to L and of that of L to M; so that K has also to M the ratio compounded of the ratios of the sides.
补全平行四边形DG。作线段K，使BC:CG = K:L，且DC:CE = L:M。
Now since, as BC is to CG, so is the parallelogram AC to the parallelogram CH, [VI. 1] while, as BC is to CG, so is K to L, therefore also, as K is to L, so is AC to CH. [V. 11] Again, since, as DC is to CE, so is the parallelogram CH to CF, [VI. 1] while, as DC is to CE, so is L to M, therefore also, as L is to M, so is the parallelogram CH to the parallelogram CF.
由VI.1，BC:CG = 平行四边形AC:CH，且DC:CE = CH:CF；结合比例传递得K:L = AC:CH，L:M = CH:CF。
[V. 11] Since then it was proved that, as K is to L, so is the parallelogram AC to the parallelogram CH, and, as L is to M, so is the parallelogram CH to the parallelogram CF, therefore, ex aequali, as K is to M, so is AC to the parallelogram CF. But K has to M the ratio compounded of the ratios of the sides; therefore AC also has to CF the ratio compounded of the ratios of the sides.
由等比定理，K:M = AC:CF。而K:M是两边比BC:CG与DC:CE的复合比，故AC:CF亦为复合比。