Proposition 12

To set up a straight line at right angles to a given plane from a given point in it.

Let the plane of reference be the given plane and A the point in it.

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It is required to set up from the point A a straight line at right angles to the plane of reference.

XI.11, I.31

From an elevated point B draw BC perpendicular to the plane of reference, and draw AD parallel to to BC through the point A.

XI.8

Then since AD and BC are two parallel straight lines, and 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.

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

Q.E.F.

Guide

This proposition, like the last, is used frequently in the rest of the Elements to construct lines perpendicular to planes.