Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

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.

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.

Therefore from the given elevated point A the straight line AF has been drawn perpendicular to the plane of reference. Q. E. F.

PROPOSITION 12.

Let the plane of reference be the given plane, and A the point in it; thus it is required to set up from the point A a straight line at right angles to the plane of reference.

Let any elevated point B be conceived, from B let BC be drawn perpendicular to the plane of reference, [XI. 11] and through the point A let AD be drawn parallel to BC. [I. 31]

Then, since AD, CB are two parallel straight lines, while one of them, BC, is at right angles to the plane of reference, therefore the remaining one, AD, is also at right angles to the plane of reference. [XI. 8]

Therefore AD has been set up at right angles to the given plane from the point A in it.