第11卷命题 18 · 直线垂直平面则过其平面皆垂直

For let any straight line AB be at right angles to the plane of reference; I say that all the planes through AB are also at right angles to the plane of reference. For let the plane DE be drawn through AB, let CE be the common section of the plane DE and the plane of reference, let a point F be taken at random on CE, and from F let FG be drawn in the plane DE at right angles to CE.

设直线AB垂直于参考平面，过AB作平面DE，设CE为平面DE与参考平面的交线。

[I. 11] Now, since AB is at right angles to the plane of reference, AB is also at right angles to all the straight lines which meet it and are in the plane of reference; [XI. Def. 3] so that it is also at right angles to CE; therefore the angle ABF is right. But the angle GFB is also right; therefore AB is parallel to FG.

在CE上任取一点F，在平面DE内作FG垂直于CE。

[I. 28] But AB is at right angles to the plane of reference; therefore FG is also at right angles to the plane of reference. [XI. 8] Now a plane is at right angles to a plane, when the straight lines drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining plane.

因为AB垂直于参考平面，所以AB垂直于CE，故角ABF为直角；又角GFB为直角，所以AB平行于FG。

[XI. Def. 4] And FG, drawn in one of the planes DE at right angles to CE, the common section of the planes, was proved to be at right angles to the plane of reference; therefore the plane DE is at right angles to the plane of reference. Similarly also it can be proved that all the planes through AB are at right angles to the plane of reference.

由于AB垂直于参考平面，故FG也垂直于参考平面；而FG在平面DE内且垂直于交线CE，因此平面DE垂直于参考平面。同理可证所有过AB的平面都垂直于参考平面。