Question

Prove that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.

Answer 1

See Fig.

ABC is the triangle and AD is the median through A. If AD be produced to a length DG = AD, then ACGB is a parallelogram.

Hence, by the parallelogram law of addition of two vectors,

But the LHS is such that vector(AB + BA)  = vector(AB - AB) = 0

Similarly, the other two pairs also become zero. Hence,

vector(AD + BE + CF) = 0