If from a parallelogram there be taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, it is about the same diameter with the whole For from the parallelogram ABCD let there be taken away the parallelogram AF similar and similarly situated to ABCD, and having the angle DAB common with it; I say that ABCD is about the same diameter with AF.
如果从一个平行四边形中截去一个与整体相似且相似放置、并与之共角的平行四边形，则截去的平行四边形与整体平行四边形共对角线。
从平行四边形 ABCD 减去共角 A 的相似平行四边形 AEFG，则两图形共享同一对角线，即 A、F、C 三点共线；H、K 为反证中辅助线交点。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
For suppose it is not, but, if possible, let AHC be the diameter < of ABCD >, let GF be produced and carried through to H, and let HK be drawn through H parallel to either of the straight lines AD, BC. [I. 31] Since, then, ABCD is about the same diameter with KG, therefore, as DA is to AB, so is GA to AK.
设平行四边形ABCD中截去平行四边形AF，AF与ABCD相似且相似放置，并共角DAB。
[VI. 24] But also, because of the similarity of ABCD, EG, as DA is to AB, so is GA to AE; therefore also, as GA is to AK, so is GA to AE.
假设ABCD与AF不共对角线，设AHC为ABCD的对角线，延长GF交AHC于H，过H作HK平行于AD。
[V. 11] Therefore GA has the same ratio to each of the straight lines AK, AE. Therefore AE is equal to AK [V. 9], the less to the greater : which is impossible.
由于ABCD与平行四边形KG共对角线，故DA比AB等于GA比AK；又由ABCD与EG相似，得DA比AB等于GA比AE。
Therefore ABCD cannot but be about the same diameter with AF; therefore the parallelogram ABCD is about the same diameter with the parallelogram AF.
因此GA比AK等于GA比AE，故AE等于AK，小等于大，矛盾。所以ABCD与AF共对角线。