The diagonals $AC$ and $BD$ of the quadrilateral $ABCD$ intersect at $O$. We are given a line $p$ that passes through $O$ and intersects $AB$ and $CD$ at $M$ and $N$, respectively; a line $q$ that passes through $M$ ($q$ is parallel to $CD$) and $q$ intersects $AC$ at $E$; a line $k$ that passes through $N$ ($k$ is parallel to $AB$) and $k$ intersects $BD$ at $F$. I should show that $BE$ is parallel to $CF$.
I showed $\dfrac{BO}{OF}=\dfrac{EO}{CO}$. Is this enough to say $BE \parallel CF$ only by using Intercept theorem (Thales' theorem)? Must I also show that $ \dfrac{OE}{OC}=\dfrac{OB}{OF}=\dfrac{EB}{CF}$? We haven't studied similar triangles!