High school Geometry

Question

Let ABCD be a trapezium. AB is parallel CD. EF is a line parallel to AB and CD. E is on BC. F is on AD. What is the length of EF in termsof sides of trapezium? Given AC/CD=M/N=BF/FC. I derived the answer for special case in which the line joins the mid points of non parallel sides. It's mean of length of parallel sides in that case. I also know that length of line segment joining the mid points of diagonal of a trapezium is mean of difference of parallel sides. Does it help? Length of parallel sides is given.

Answer 1

Let $ABCD$ be our trapezoid, where $BC||AD$, $AD=a$, $BC=b$, where $a>b$.

$M\in AB$ and $N\in CD$ such that $MN||AD$, $MN=x$ and $AM:MB=n:m$.

Let $K\in AD$ such that $BK||CD$. Also, let $BK\cap MN=\{L\}$.

Thus, $KD=LN=BC=b$ and $$\frac{ML}{AK}=\frac{BM}{AB}$$ or $$\frac{x-b}{a-b}=\frac{m}{m+n}$$ or $$x=\frac{ma+nb}{m+n}.$$ Done!