Book 11
5
If a straight line be set up at right angles to three straight lines which meet one another, at their common point of section, the three straight lines are in one plane.
5
For let a straight line AB be set up at right angles to the three straight lines BC, BD, BE, at their point of meeting at B; I say that BC, BD, BE are in one plane.
5
For suppose they are not, but, if possible, let BD, BE be in the plane of reference and BC in one more elevated; let the plane through AB, BC be produced; it will thus make, as common section in the plane of reference, a straight line. [XI. 3]
5
Let it make BF.
5
Therefore the three straight lines AB, BC, BF are in one plane, namely that drawn through AB, BC.
5
Now, since AB is at right angles to each of the straight lines BD, BE, therefore AB is also at right angles to the plane through BD, BE. [XI. 4]
5
But the plane through BD, BE is the plane of reference; therefore AB is at right angles to the plane of reference.
5
Thus AB will also make right angles with all the straight lines which meet it and are in the plane of reference. [XI. Def. 3]
5
But BF which is in the plane of reference meets it; therefore the angle ABF is right.
5
But, by hypothesis, the angle ABC is also right; therefore the angle ABF is equal to the angle ABC.
5
And they are in one plane: which is impossible.
5
Therefore the straight line BC is not in a more elevated plane; therefore the three straight lines BC, BD, BE are in one plane.
5
Therefore, if a straight line be set up at right angles to three straight lines, at their point of meeting, the three straight lines are in one plane. Q. E. D.
PROPOSITION 6.
6
If two straight lines be at right angles to the same plane, the straight lines will be parallel.
6
For let the two straight lines AB, CD be at right angles to the plane of reference; I say that AB is parallel to CD.
