| 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. | XI.16 | |
| 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. | VI.2 | |
| But it was prove that AO is to OD as AE is to EB, therefore AE is to EB as CF is to FD. | V.11 | |
| Therefore, if two straight lines are cut by parallel planes, then they are cut in the same ratios. | ||
| Q. E. D. | ||
Book XI Introduction - Proposition XI.16 - Proposition XI.18.