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.
be two parallel straight lines, and let points E
be taken at random on them respectively; I say that the straight line joining the points E
is in the same plane with the parallel straight lines.
For suppose it is not, but, if possible, let it be in a more elevated plane as EGF
, and let a plane be drawn through EGF
; it will then make, as section in the plane of reference, a straight line. [XI. 3
Let it make it, as EF
; therefore the two straight lines EGF
will enclose an area: which is impossible.
Therefore the straight line joined from E
is not in a plane more elevated; therefore the straight line joined from E
is in the plane through the parallel straight lines AB
Therefore etc. Q. E. D.