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 $C$ be 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 $CD$; [Prop. 1.3]on $DE$ let the equilateral triangle $FDE$ be constructed, [Prop. 1.1]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 two sides $DC$, $CF$ are equal to the two sides $EC$, $CF$ respectively;and the base $DF$ is equal to the base $FE$;therefore the angle $DCF$ is equal to the angle $ECF$; [Prop. 1.8]and they are adjacent angles.But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right; [Def. 10]therefore each of the angles $DCF$, $FCE$ is right.Therefore the straight line $CF$ has been drawn at right angles to the given straight line $AB$ from the given point $C$ on it.Q.E.F.