|If two parallel planes are cut by any plane, then their intersections are parallel.|
|Let the two parallel planes AB and CD be cut by the plane EFGH, and let EF and GH be their intersections.||XI.3|
|I say that EF is parallel to GH.|
|If not, then EF and GH will, when produced, meet either in the direction of F and H or in the direction of E and G.
First, let them meet when produced in the direction of F and H at K.
|Now, since EFK lies in the plane AB, therefore all the points on EFK also lie in the plane AB. But K is one of the points on the straight line EFK, therefore K lies in the plane AB. For the same reason K also lies in the plane CD. Therefore the planes AB and CD will meet when produced.||XI.1|
|But they do not meet, because, by hypothesis, they are parallel. Therefore the straight lines EF and GH do not meet when produced in the direction of F and H.|
|Similarly we can prove that neither do the straight lines EF and GH meet when produced in the direction of E and G.|
|But straight lines which do not meet in either direction are parallel. Therefore EF is parallel to GH.|
|Therefore, if two parallel planes are cut by any plane, then their intersections are parallel.|
|Q. E. D.|
Next proposition: XI.17