In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another.
在任意平行四边形中，关于对角线的两个平行四边形既与整个平行四边形相似，也彼此相似。
平行四边形 ABCD 的对角线 AC 上取 K；过 K 作平行于 AB、AD 的直线把 ABCD 分成四块，其中位于对角线两旁的 EG、FH 两个小平行四边形彼此相似且都与 ABCD 相似。
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Let ABCD be a parallelogram, and AC its diameter, and let EG, HK be parallelograms about AC; I say that each of the parallelograms EG, HK is similar both to the whole ABCD and to the other. For, since EF has been drawn parallel to BC, one of the sides of the triangle ABC, proportionally, as BE is to EA, so is CF to FA. [VI. 2] Again, since FG has been drawn parallel to CD, one of the sides of the triangle ACD, proportionally, as CF is to FA, so is DG to GA.
设ABCD为平行四边形，AC为其对角线，EG和HK是关于AC的平行四边形。需证EG和HK各与ABCD相似，且彼此相似。
[VI. 2] But it was proved that, as CF is to FA, so also is BE to EA; therefore also, as BE is to EA, so is DG to GA, and therefore, componendo, as BA is to AE, so is DA to AG, [V. 18] and, alternately, as BA is to AD, so is EA to AG. [V. 16] Therefore in the parallelograms ABCD, EG, the sides about the common angle BAD are proportional. And, since GF is parallel to DC, the angle AFG is equal to the angle DCA; and the angle DAC is common to the two triangles ADC, AGF; therefore the triangle ADC is equiangular with the triangle AGF.
因EF平行于BC，由三角形ABC得BE比EA等于CF比FA；又因FG平行于CD，由三角形ACD得CF比FA等于DG比GA，故BE比EA等于DG比GA。
For the same reason the triangle ACB is also equiangular with the triangle AFE, and the whole parallelogram ABCD is equiangular with the parallelogram EG. Therefore, proportionally, as AD is to DC, so is AG to GF, as DC is to CA, so is GF to FA, as AC is to CB, so is AF to FE, and further, as CB is to BA, so is FE to EA. And, since it was proved that, as DC is to CA, so is GF to FA, and, as AC is to CB, so is AF to FE, therefore, ex aequali, as DC is to CB, so is GF to FE.
合比得BA比AE等于DA比AG，更比得BA比AD等于EA比AG，因此平行四边形ABCD与EG的边关于公共角BAD成比例。
[V. 22] Therefore in the parallelograms ABCD, EG the sides about the equal angles are proportional; therefore the parallelogram ABCD is similar to the parallelogram EG. [VI. Def. 1] For the same reason the parallelogram ABCD is also similar to the parallelogram KH; therefore each of the parallelograms EG, HK is similar to ABCD. But figures similar to the same rectilineal figure are also similar to one another; [VI. 21] therefore the parallelogram EG is also similar to the parallelogram HK.
由平行线得角相等，可证三角形相似，进而各边比例成立，故ABCD与EG相似；同理ABCD与HK相似，且相似于同一图形的两图形彼此相似，故EG与HK相似。