G.515 point segments set

XI.13

From the same point two segments cannot be set up at right angles to the same plane on the same side.

point_segments_set()

prove They are in one plane ⇒⇐

construct If possible, from the same point A let / A B /, / A C / be set up at right angles to the plane of reference and on the same side, and let a plane be drawn through / B A /, / A C /

∵ planes_cut_commonassert It will then make, as section through A in the plane of reference, a segment

assert Let it make / D A E /

assert / A B /, / A C /, / D A E / are in one plane

∵ segment_right_angles_defassert since / C A / is at right angles to the plane of reference, it will also make right angles with all the segments which meet it and are in the plane of reference

assert / D A E / meets it and is in the plane of reference

assert < C A E ) is right

assert For the same reason < B A E ) is also right

assert < C A E ) = < B A E )

∴ They are in one plane ⇒⇐

Dependency Graph

Required for

G.523