In a triangle $O A B, E$ is the midpoint of $B O$ and $D$ is a point on $A B$. Such that $A D: D B=2

Question

In a triangle $O A B, E$ is the midpoint of $B O$ and $D$ is a point on $A B$. Such that $A D: D B=2: 1$. If $O D$ and $A E$ intersect at $P$, if the ratio $O P: P D=m:n$ in the lowest form, find $m^2+n^2$.\

Professor PC Thomas Classes · Answer

With $O$ as the origin of reference let $\vec{a}, \vec{b}$ be the position vectors of $A$ and $B$ respectively.\ Position vector of $D$ is $\frac{\vec{a}+2 \vec{b}}{3}$\ Position vector of $E$ is $\frac{\vec{b}}{2}$\ Equations of $O D$ and $A E$ are