Question

$ABCD$ is quadrilateral $E,F,G$ and $H$ are the midpoints of $AB,BC,CD$ and $DA$ respectively. Prove that $EFGH$ is a parallelogram

Answer 1

Given $ABCD$ is a quadrilateral $E,F,G,H$ are the midpoint of $AB,BC,CD,DA$ respectively

In $\Delta ADC$

H and G the mid point of AD and DC

$\therefore GH\parallel AC$

And $GH=\frac{1}{2}AC$

In $\Delta ABC$

$E$ and $F$ the mid point of $AB$ and $BC$

$\therefore EF\parallel AC$

And $EF=\frac{1}{2}AC$

So $EF=GH$ and $EFIIGH$

So the quadrilateral $EFGH$

Opposite sides are parallel and equal

Then $EFGH$ is a parallelogram