G.509 segments parallel points

XI.7

If two segments be parallel and points be taken at random on each of them, the segment joining the points is in the same plane with the parallel lines.
segments_parallel_points()
prove the segment joining E, F is in the same plane with the parallel lines
A E B G C F D
construct Let / A B /, / C D / be two parallel lines, and let points E, F be taken at random on them respectively
construct Suppose it is not, but, if possible, let it be in a more elevated plane as ( E G F ), and let a plane be drawn through ( E G F )
∵ planes_cut_commonassert It will then make, as section in the plane of reference, a segment
assert Let it make it, as / E F /
assert ( E G F ), / E F / will enclose an area ⇒⇐
∴ the segment joined from E to F is not in a plane more elevated; therefore the segment joined from E to F is in the plane through the parallel lines / A B /, / C D /

Dependency Graph

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Required for

G.510, G.511, G.512, G.513, G.514, G.518, G.522, G.528, G.530, G.531, G.532, G.541, G.542, G.543, G.544, G.545, G.546, G.547, G.549, G.550, G.554, G.555, G.556, G.557, G.558, G.559, G.560, G.563, G.565, G.570, G.571, G.572, G.573, G.577, G.578, G.592, G.594, G.600, G.601