Question

Prove that the line segments joining the middle points of the sides of a triangle divide it into four congruent triangles.

Answer 1

As, D and E are the mid points of sides AB, and BC of $△ABC$
$\therefore DE\parallel AC\left(Bymidpointtheorem\right)$
Similarly,$DF\parallel BCandEF\parallel AB$
Therefore, ADEF, BDFE and DFCE are all parallelograms.
Now, DE is the diagonal of the parallelogram BDFE.
$\therefore △BDE\cong △FED$
Similarly DF is the ​ diagonal of the parallelogram ADEF.
$\therefore △DAF\cong △FED$
And, EF is the ​ diagonal of the parallelogram DFCE.
$\therefore △\cong △FED$
So, all the four triangles are congruent.