1.13: Past-paper question 10

The points $\displaystyle A$, $\displaystyle B$, $\displaystyle C$ and $\displaystyle D$ are the vertices of a quadrilateral such that

(a) Show that $\displaystyle ABCD$ is a parallelogram.

$\displaystyle BC$ is extended to the point $\displaystyle E$ such that $\displaystyle BCE$ is a straight line.

Point $\displaystyle F$ lies on $\displaystyle CD$ such that $\displaystyle CF : FD = 1 : 2$

Given that $\displaystyle A$, $\displaystyle F$ and $\displaystyle E$ are collinear,

(b) find the vector $\displaystyle \overrightarrow{AE}$ in the form $\displaystyle X\mathbf{a} + Y\mathbf{b}$ where $\displaystyle X$ and $\displaystyle Y$ are rational numbers to be found.