If two straight lines are cut by parallel planes, then they are cut in the same ratios.

Let the two straight lines *AB* and *CD* be cut by the parallel planes *GH, KL,* and *MN* at the points *A, E,* and *B,* and at the points *C, F,* and *D,* respectively.

I say that the straight line *AE* is to *EB* as *CF* is to *FD.*

Join *AC, BD,* and *AD.* Let *AD* meet the plane *KL* at the point *O.* Join *EO* and *FO.*

Now, since the two parallel planes *KL* and *MN* are cut by the plane *EBDO,* therefore their intersections *EO* and *BD* are parallel. For the same reason, since the two parallel planes *GH* and *KL* are cut by the plane *AOFC,* their intersections *AC* and *OF* are parallel.

And, since the straight line *EO* is parallel to *BC,* one of the sides of the triangle *ABD,* therefore proportionally *AE* is to *EB* as *AO* is to *OD.* Again, since the straight line *FO* is parallel to *CA,* one of the sides of the triangle *ADC,* therefore proportionally *AO* is to *OD* as *CF* is to *FD.*

But it was prove that *AO* is to *OD* as *AE* is to *EB,* therefore *AE* is to *EB* as *CF* is to *FD.*

Therefore, *if two straight lines are cut by parallel planes, then they are cut in the same ratios.*

Q.E.D.