To bisect a given rectilineal angle.

Let the angle $BAC$ be the given rectilineal angle.Thus it is required to bisect it.Let a point $D$ be taken at random on $AB$;let $AE$ be cut of from $AC$ equal to $AD$; [Prop. 1.3]let $DE$ be joined, and on $DE$ let the equilateral triangle $DEF$ be constructed;let $AF$ be joined.I say that the angle $BAC$ has been bisected by the straight line $AF$.For since $AD$ is equal to $AE$ and $AF$ is common,the two sides $DA$, $AF$ are equal to the two sides $EA$, $AF$ respectively.And the base $DF$ is equal to the base $EF$;therefore the angle $DAF$ is equal to the angle $EAF$. [Prop. 1.8]Therefore the given rectilineal angle $BAC$ has been bisected by the straight line $AF$.Q.E.F.