Question

If the midpoints of the sides of a quadrilateral are joined in an order (in succession), prove that the resulting quadrilateral is a parallelogram.

Answer 1

Given: Quadrilateral $ABCD$ with $E$, $F$, $G$, $H$ to be the mid-points of $AB$, $BC$, $CD$, $DA$ respectively.
To prove: $EFGH$ is a parallelogram.
Construction: Draw the diagonal $AC$.
Proof: Look at the triangle $ADC$. We know that
$H$ is the mid-point of $AD$ and $G$ is the mid-point of $DC$.
By midpoint theorem $HG\parallel AC$. Similarly, in triangle $ABC$, we know that
$E$ is the mid-point of $AB$ and $F$ is the mid-point of $BC$.
Again by mid-point theorem, $EF\parallel AC$. It follows that
$HG\parallel EF$.
We also know
$GH=\frac{AC}{2}=EF$.
The quadrilateral $EFGH$ is such that $EF=GH$ and $EF\parallel GH$. Thus by theorem 2, $EFGH$ is a parallelogram.