Construction: External Angle Bisector

This construction is exactly the same as that of the internal angle bisector.

To construct an angle bisector in general, not just an angle bisector, first consider ray $\overrightarrow{LI}$ and ray $\overrightarrow{MI}$, intersecting at $I$. Using a compass, we draw a circle centered at point $I$ with any radius you want which intersects $\overrightarrow{LI}$ and $\overrightarrow{MI}$ at $X$ and $Y$, respectively (this is so that $IX=IY$). Now we draw two circles centered at $X$ and $Y$ respectively such that the two circles have the same radius, and the two circles intersect at $Z$. $\overline{IZ}$ is the angle bisector of $\angle LIM$.

Basically, construct points $X$ and $Y$ on $\overrightarrow{LI}$ and $\overrightarrow{MI}$ such that $IX=IY$, then construct another point $Z$ such that $XZ=YZ$.

Why does this construction work? Symmetry!