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 AB
be set up at right angles to the plane of reference and on the same side, and let a plane be drawn through BA
; it will then make, as section through A
in the plane of reference, a straight line. [XI. 3
Let it make DAE
; therefore the straight lines AB
are in one plane.
And, since CA
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. [XI. Def. 3
meets it and is in the plane of reference; therefore the angle CAE
For the same reason the angle BAE
is also right; therefore the angle CAE
is equal to the angle BAE
And they are in one plane: which is impossible.
Therefore etc. Q. E. D.