Q1 of 177 Page 23

If O is a point in space, ABC is a triangle and D, E, F are the mid-points of the sides BC, CA and AB respectively of the triangle, prove that

Let position vectors of the vertices A, B and C of ΔABC with respect to O be , and respectively.


⇒


Let us also assume the position vectors of the midpoints D, E and F with respect to O are, and respectively.


⇒



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 and respectively, internally in the ratio m : n is



Here, m = n = 1





Similarly, for midpoint E and side CA, we get and for midpoint F and side AB, we get .


Adding these three equations, we get






Thus, .


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