therefore GH is at right angles to the plane passing through ED and DA; [XI. 8.
therefore GH makes right angles with every straight line meeting it in that plane. [XI. Definition 3.
But AF meets it, and is in the plane passing through ED and DA;
therefore GH is at right angles to AF,
and therefore AF is at right angles to GH.
But is also at right angles to DE; [Construction.
therefore AF is at right angles to each of the straight lines GH and DE at the point of their intersection;
therefore AF is perpendicular to the plane passing through GH and DE, that is, to the plane BH. [XI. 4.
Wherefore, from the given point A, without the plane BH, the straight line AF has been drawn perpendicular to the plane. q.e.f.
PROPOSITION 12. PROBLEM.
To erect a straight line at right angles to a given plane, from a given point in the plane.
Let A be the given point in the given plane: it is required to erect a straight line from the point A, at right angles to the plane.
From any point B without the plane, draw BC perpendicular to the plane; [XI. 11.
and from the point A draw AD parallel to BC, [I. 31.
AD shall be the straight line required.
For, because AD and BC are two parallel straight lines, [Constr.
and that one of them BC is at right angles to the given plane, [Construction.
the other AD is also at right angles to the given plane. [XI. 8,
Wherefore a straight line has been erected at right an-gles to a given plane, from a given point in it. q.e.f.
