G.515 point segments set — Heath

XI.13

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

For, if possible, from the same point A let the two straight lines / 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 /; it will then make, as section through A in the plane of reference, a straight line. [G.504]

Let it make / D A E /;

therefore the straight lines / A B /, / A C /, / D A E / are in one plane.

And, since / C A / is at right angles to the plane of reference, it will also make right angles with all the straight lines which meet it and are in the plane of reference. [G.505]

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

therefore the angle < C A E ) is right.

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

therefore the angle < C A E ) is equal to the angle < B A E ).

And they are in one plane: which is impossible.

Therefore etc. Q. E. D.

Dependency Graph

Required for

G.523