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Proposition 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 AB and AC be set up at right angles to the plane of reference and on the same side.
| Draw a plane through BA and AC. It intersects the plane of reference in a straight line through A. Let the line be DAE. | XI.3
| Therefore the straight lines AB, AC, and DAE lie in one plane. And, since CA is at right angles to the plane of reference, it also makes right angles with all the straight lines which meet it and lie in the plane of reference. | XI.Def.3 | But DAE meets it and lies in the plane of reference, therefore the angle CAE is right. For the same reason the angle BAE is also right. Therefore the angle CAE equals the angle BAE. | And they lie in one plane, which is impossible.
Therefore, from the same point two straight lines cannot be set up at right angles to the same plane on the same side.
Q. E. D.

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This proposition is used in the proof of proposition XI.19.

Book XI Introduction - Proposition XI.12 - Proposition XI.14.

© 1996
D.E.Joyce
Clark University