| If two parallel planes are cut by any plane, then their intersections are parallel. | ||
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Let the two parallel planes AB and CD be cut by the plane EFGH, and let EF and GH be their intersections.
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. |
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| 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. | ||
Book XI Introduction - Proposition XI.15 - Proposition XI.17.