Median of Trapezoid is Parallel to Bases/Necessary Condition
Theorem
Let $\Box ABCD$ be a trapezoid such that $AB$ and $DC$ are the parallel sides.
Let $E$ be the midpoint of $AD$.
Let $F$ be the midpoint of $BC$.
Then $EF$ is parallel to both $AB$ and $DC$.
Proof
Aiming for a contradiction, suppose $EF$ is not parallel to $DC$.
By Playfair's axiom, there exists a unique straight line through $E$ which is parallel to $DC$.
Let $EF'$ be this line.
From Median of Trapezoid is Parallel to Bases: Sufficient Condition, $F'$ is the midpoint of $BC$.
But by hypothesis $F$ is also the midpoint of $BC$.
That is:
- $F = F'$
So:
- $EF = EF'$
and so $EF$ is parallel to $DC$.
This contradicts our assertion that $EF$ is not parallel to $DC$.
Hence, by Proof by Contradiction, $EF$ is parallel to $DC$.
By definition of base of trapezoid, $AB$ is parallel to $DC$.
By Parallelism is Transitive Relation, $EF$ is parallel to $AB$.
That is, $EF$ is parallel to both $AB$ and $DC$.
$\blacksquare$