Question: Let $AOB$ be a given angle less than $180^\circ$ and let $P$ be an interior point of the angular region of $\angle AOB$. Show, with proof, how to construct, using only ruler and compass, a line segment $CD$ passing through $P$ such that $C$ lies on ray $OA$ and $D$ lies on ray $OB$, and $CP : PD=1 : 2$.
My attempt: I thought that constructing a triangle and constructing medians for every side would give me the centroid, which divides the median in the ratio $2:1$.
But how can I construct a triangle?