If a straight line be set up at right angles to three straight lines which meet one another, at their common point of section, the three straight lines are in one plane.
如果一条直线垂直于三条相交于同一点的直线，那么这三条直线共面。
For let a straight line AB be set up at right angles to the three straight lines BC, BD, BE, at their point of meeting at B; I say that BC, BD, BE are in one plane. For suppose they are not, but, if possible, let BD, BE be in the plane of reference and BC in one more elevated; let the plane through AB, BC be produced; it will thus make, as common section in the plane of reference, a straight line. [XI. 3] Let it make BF.
设直线AB垂直于三条相交于B的直线BC、BD、BE。假设BD、BE在参考平面内，而BC在另一平面内。
Therefore the three straight lines AB, BC, BF are in one plane, namely that drawn through AB, BC. Now, since AB is at right angles to each of the straight lines BD, BE, therefore AB is also at right angles to the plane through BD, BE. [XI. 4] But the plane through BD, BE is the plane of reference; therefore AB is at right angles to the plane of reference.
过AB、BC作平面，与参考平面交于直线BF。则AB、BC、BF共面。
Thus AB will also make right angles with all the straight lines which meet it and are in the plane of reference. [XI. Def. 3] But BF which is in the plane of reference meets it; therefore the angle ABF is right. But, by hypothesis, the angle ABC is also right; therefore the angle ABF is equal to the angle ABC.
由于AB垂直于BD、BE，故AB垂直于BD、BE所在平面（即参考平面），从而AB垂直于参考平面内所有过B的直线，包括BF，故角ABF为直角。
And they are in one plane: which is impossible. Therefore the straight line BC is not in a more elevated plane; therefore the three straight lines BC, BD, BE are in one plane.
但已知角ABC也是直角，因此角ABF等于角ABC，且在同一平面内，矛盾。故BC不在另一平面内，即BC、BD、BE共面。