构造：内角平分线

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!