Vector Method?

Question

How to Prove the line joining the midpoints of the sides of the equilateral triangle taken in order form a parallelogram using vector method.

Answer 1

They don't. They form another equilateral triangle. An equilateral triangle has three sides, and therefore three midpoints. If you join three points you get three sides. That's a triangle, not a parallelogram.

Answer 2

I understand your question as under.
In ΔABC, D, E and F are midpoints of sides AB, BC and CA respectively.
To prove vectorially that quadrlateral ADEF is a parallelogram.

Let AB and AC be b and c vectors w.r.t. A as origin.
Then AD = b/2 vector
AF = c/2 vector
AE = (1/2) (b + c) vector

We have to prove that the lengths any pair of opposite sides of the quadrilateral ADEF are equal and parallel.

Let us consider sides AD and FE
l AD l = l b/2 vectorl
l FE l = l AE vector - AF vector l
= l(1/2)(b + c) - (1/2)c l = l b/2 vector l
Thus, lengths Ad and FE are equal.
Moreover, AD vector = b/2 vector = FE vector. So, they are parallel.
Thus. quadrilateral ADFE is a parallelogram.