第11卷命题 11 · 过定点作平面垂线

Let A be the given elevated point, and the plane of reference the given plane; thus it is required to draw from the point A a straight line perpendicular to the plane of reference. Let any straight line BC be drawn, at random, in the plane of reference, and let AD be drawn from the point A perpendicular to BC. [I. 12] If then AD is also perpendicular to the plane of reference, that which was enjoined will have been done.

在给定平面内任取一直线BC，从点A作AD垂直于BC。若AD垂直于平面，则完成。

But, if not, let DE be drawn from the point D at right angles to BC and in the plane of reference, [I. 11] let AF be drawn from A perpendicular to DE, [I. 12] and let GH be drawn through the point F parallel to BC. [I. 31] Now, since BC is at right angles to each of the straight lines DA, DE, therefore BC is also at right angles to the plane through ED, DA. [XI. 4] And GH is parallel to it; but, if two straight lines be parallel, and one of them be at right angles to any plane, the remaining one will also be at right angles to the same plane; [XI. 8] therefore GH is also at right angles to the plane through ED, DA.

否则，在平面内过D作DE垂直于BC，再从A作AF垂直于DE。

Therefore GH is also at right angles to all the straight lines which meet it and are in the plane through ED, DA. [XI. Def. 3] But AF meets it and is in the plane through ED, DA; therefore GH is at right angles to FA, so that FA is also at right angles to GH. But AF is also at right angles to DE; therefore AF is at right angles to each of the straight lines GH, DE.

过F作GH平行于BC。因BC垂直于DA和DE，故BC垂直于它们确定的平面；GH平行于BC，故GH也垂直于该平面。

But, if a straight line be set up at right angles to two straight lines which cut one another, at the point of section, it will also be at right angles to the plane through them; [XI. 4] therefore FA is at right angles to the plane through ED, GH. But the plane through ED, GH is the plane of reference; therefore AF is at right angles to the plane of reference.

GH垂直于平面内过F的直线AF，且AF垂直于DE，故AF垂直于相交直线GH和DE，从而垂直于它们确定的平面，即给定平面。