If two straight lines be cut by parallel planes, they will be cut in the same ratios.
若两条直线被平行平面所截，则它们被截成的线段成比例。
For let the two straight lines AB, CD be cut by the parallel planes GH, KL, MN at the points A, E, B and C, F, D; I say that, as the straight line AE is to EB, so is CF to FD. For let AC, BD, AD be joined, let AD meet the plane KL at the point O, and let EO, OF be joined.
连接AC、BD、AD，设AD与平面KL交于点O，连接EO、OF。
Now, since the two parallel planes KL, MN are cut by the plane EBDO, their common sections EO, BD are parallel. [XI. 16] For the same reason, since the two parallel planes GH, KL are cut by the plane AOFC, their common sections AC, OF are parallel.
由于平行平面KL、MN被平面EBDO所截，其交线EO、BD平行；同理，平行平面GH、KL被平面AOFC所截，其交线AC、OF平行。
[id.] And, since the straight line EO has been drawn parallel to BD, one of the sides of the triangle ABD, therefore, proportionally, as AE is to EB, so is AO to OD. [VI. 2] Again, since the straight line OF has been drawn parallel to AC, one of the sides of the triangle ADC, proportionally, as AO is to OD, so is CF to FD.
在三角形ABD中，EO平行于BD，故AE比EB等于AO比OD。
[id.] But it was also proved that, as AO is to OD, so is AE to EB; therefore also, as AE is to EB, so is CF to FD.
在三角形ADC中，OF平行于AC，故AO比OD等于CF比FD；因此AE比EB等于CF比FD。