Let position vectors of the vertices A, B and C of ΔABC with respect to O be \(\vec a,\vec b\) and \(\vec c\)respectively.
⇒\(\vec{OA}=\vec a,\) \(\vec {OB}=\vec b\) and \(\vec {OC}=\vec c\)
Let us also assume the position vectors of the midpoints D, E and F with respect to O are \(\vec d, \vec e\) and \(\vec f\)respectively.
⇒ \(\vec{OD}=\vec d,\vec{OE}=\vec e\) and \(\vec OF=\vec f\)
Now, D is the midpoint of side BC.
This means D divides BC in the ratio 1 : 1.
Recall the position vector of point P which divides AB, the line joining points A and B with position vectors \(\vec a\) and \(\vec b\) respectively, internally in the ratio m : n is
Similarly, for midpoint E and side CA, we get \(\vec c+\vec a=2\vec e\) and for midpoint F and side AB, we get \(\vec a+\vec b=2\vec f\)
Adding these three equations, we get