G.509 segments parallel points — Heath

XI.7

If two straight lines be parallel and points be taken at random on each of them, the straight line joining the points is in the same plane with the parallel straight lines.

Let / A B /, / C D / be two parallel straight lines, and let points E, F be taken at random on them respectively; I say that the straight line joining the points E, F is in the same plane with the parallel straight lines.

A E B G C F D

For 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 );

it will then make, as section in the plane of reference, a straight line. [G.504]

Let it make it, as / E F /;

therefore the two straight lines ( E G F ), / E F / will enclose an area: which is impossible.
Therefore the straight line joined from E to F is not in a plane more elevated; therefore the straight line joined from E to F is in the plane through the parallel straight lines / A B /, / C D /.
Therefore etc. Q. E. D.

Dependency Graph

digraph { bgcolor="transparent"; node [shape=box, style="rounded,filled", fontname="Helvetica", fontsize=11, color="#d6d3cd", fontcolor="#ffffff"]; edge [color="#8c857b", fontcolor="#8c857b", arrowsize=0.75]; rankdir="TB"; "G.1" [label="G.1", tooltip="G.1 (I.def.1): A point A is that which has no part.", fillcolor="#224422", URL="../G.1/index.html", target="_top"]; "G.2" [label="G.2", tooltip="G.2 (I.def.2): A line [ A B ] is a projection of a relationship of two points A, B.", fillcolor="#224422", URL="../G.2/index.html", target="_top"]; "G.3" [label="G.3", tooltip="G.3 (I.def.3): The extremities of a segment / A B / are points A, B.", fillcolor="#224422", URL="../G.3/index.html", target="_top"]; "G.4" [label="G.4", tooltip="G.4 (I.def.4): A segment / A B / is a portion of a line [ A B ] between two points A, B on the line.", fillcolor="#224422", URL="../G.4/index.html", target="_top"]; "G.5" [label="G.5", tooltip="G.5 (I.def.5): A surface is that which has length and breadth only.", fillcolor="#224422", URL="../G.5/index.html", target="_top"]; "G.7" [label="G.7", tooltip="G.7 (I.def.7): A plane surface lies evenly with the lines on itself.", fillcolor="#224422", URL="../G.7/index.html", target="_top"]; "G.52" [label="G.52", tooltip="G.52 (I.def.23): Parallel lines are lines in the same plane which, extended indefinitely in both directions, do not meet.", fillcolor="#224422", URL="../G.52/index.html", target="_top"]; "G.504" [label="G.504", tooltip="G.504 (XI.3): If two planes cut one another, their common section is a segment.", fillcolor="#222244", URL="../G.504/index.html", target="_top"]; "G.509" [label="G.509", tooltip="G.509 (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.", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", URL="index.html", target="_top"]; "G.2" -> "G.1"; "G.3" -> "G.1"; "G.4" -> "G.1"; "G.509" -> "G.1"; "G.3" -> "G.2"; "G.4" -> "G.2"; "G.7" -> "G.2"; "G.52" -> "G.2"; "G.504" -> "G.2"; "G.509" -> "G.2"; "G.4" -> "G.3"; "G.504" -> "G.3"; "G.7" -> "G.4"; "G.52" -> "G.4"; "G.504" -> "G.4"; "G.509" -> "G.4"; "G.7" -> "G.5"; "G.52" -> "G.7"; "G.504" -> "G.7"; "G.509" -> "G.7"; "G.509" -> "G.52"; "G.509" -> "G.504"; }

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