Question

If $D,E,F$ are midpoint of the sides $BC,CA$ an $AB$ of triangle $ABC$, then show that $BDEF$ is a parallelogram

Answer 1

In $△ABC,$

$F$ is the mid-point of $AB$,

$E$ is the mid-point of $AC$,

$\therefore FE\parallel BC$

$\Rightarrow FE\parallel BD$ since parts of parallel lines are parallel .......$\left(1\right)$

$D$ is the mid-point of $BC$,

$E$ is the mid-point of $AC$,

$\therefore DE\parallel AB$

$\Rightarrow DE\parallel FB$ since parts of parallel lines are parallel .......$\left(2\right)$

Now, $FE\parallel BD$ and $DE\parallel FB$

In $BDEF$, both pairs of opposite sides are parallel,

$\therefore BDEF$ is a parallelogram.

Hence proved.