Question

In ΔABC, median AD is produced to E such that AD = DE, prove that ABEC is a parallelogram.

Answer 1

Given:

In triangle ABC,

AD is a median,

So, D is the midpoint of BC.

AD is produced to E such that AD = DE.

To Prove: ABEC is a parallelogram.

Proof [Step-wise]:

1. From AD being a median

D is the midpoint of BC 

⇒ BD = DC

2. Given AD = DE

D is the midpoint of AE

⇒ AD = DE

3. Therefore, D is the midpoint of both BC and AE.

So, the segments BC and AE bisect each other at D.

4. If the diagonals here BC and AE of a quadrilateral bisect each other, the quadrilateral is a parallelogram.

5. Hence, ABEC is a parallelogram since in quadrilateral ABEC, the diagonals AE and BC bisect each other.