Does the Trapezoid Midsegment theorem work the other way?

Question

Let $ABCD$ be a trapezoid with parallel bases $AB$ and $CD$. Let $E$ be the midpoint of $AC$ and $F$ be some point of $BD$. The line $EF$ is parallel to $AB$ and $CD$.

Can we prove that $F$ is the midpoint of $CD$ using vectors?

Answer 1

I am sure it can be done with vectors, but it can also easily be done with coordinate geometry. Place $A(0,0), B(a,0), C(b,c)$ and $D(d,c)$ as a trapezoid in the first quadrant. Then from the given we have $E(0.5b,0.5c)$ and since $EF$ is parallel to $AB$, it follows $F(f,0.5c)$. To be shown: $f=\frac{a+d}{2}$. Now set up an equation for the line passing through $B$ and $D$ and substitute $0.5c$ for the $y$. Solving for $x$ gives the $f$ value, which is what I want you to work out.