The Thirteen Books of the Elements, Volume 3 by Euclid

Author:Euclid
Language: eng
Format: epub
Publisher: Dover Publications
Published: 1956-05-13T04:00:00+00:00

[This proposition and the proof are in Legendre.]

The following theorems follow as corollaries.

2. Through a given straight line a plane can be drawn parallel to any other given straight line; and, if the lines are not parallel, only one such plane can be drawn.

We have simply to draw through any point on the first line a straight line parallel to the second line and then pass a plane through these two intersecting lines. This plane is then, by the above proposition, parallel to the second given straight line.

3. Through a given point a plane can be drawn parallel to any two straight lines in space; and, if the latter are not parallel, only one such plane can be drawn.

Here we draw through the point straight lines parallel respectively to the given straight lines and then draw a plane through the lines so drawn.

Next we have the partial converse of the first proposition above.

4. If a straight line is parallel to a plane, it is also parallel to the intersection of any plane through it with the given plane.

Let AB be parallel to the plane MN, and let any plane through AB intersect MN in CD.

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