Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/250

226 PROPOSITION 5. THEOREM.

If three straight lines meet all at one point, and a straight line stand at right angles to each of them at that point, the three straight lines shall he in one and the same plane.

Let the straight line AB stand at right angles to each of the straight lines BC, BD, BE, at B the point where they meet: BC, BD, BE shall be in one and the same plane.

For, if not, let, if possible, BD and BE be in one plane, and BC without it; let a plane pass through AB and BC; the common section of this plane with the plane in which are BD and BE is a straight line; [XI. 3.

let this straight line be BF.

Then the three straight lines AB, BC, BF are all in one plane, namely, the plane which passes through AB and BC.

And because AB stands at right angles to each of the straight lines BD, BE, [Hypothesis.

therefore it is at right angles to the plane passing through them; [XI. 4.

therefore it makes right angles with every straight line meeting it in that plane. [XI. Definition 3.

But BF meets it, and is in that plane;

therefore the angle ABF is a right angle.

But the angle ABC is, by hypothesis, a right angle;

therefore the angle ABC is equal to the angle ABF; [Ax. 11.

and they are in one plane; which is impossible. [Axiom 9.

Therefore the straight line BC is not without the plane in which are BD and BE,

therefore the three straight lines BC, BD, BE are in one and the same plane.

Wherefore, if three straight lines &c.