Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/257

Rh 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.

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.