To draw a straight line at right angles to a given straight line from a given point on it.
Let AB be the given straight line, and C the given point on it.
Thus it is required to draw from the point C a straight line at right angles to the straight line AB. Let a point D be taken at random on AC;
let CE be made equal to CD; [I. 3] on DE let the equilateral triangle FDE be constructed, [I. 1] and let FC be joined; I say that the straight line FC has been drawn at right
angles to the given straight line AB from C the given point on it. For, since DC is equal to CE, and CF is common,
the adjacent angles equal to one another, each of the equal angles is right; [Def. 10]
C on it.
Q. E. F. 1