Mastering Mathematics Smartly
by Wee Wen Shih

Question:

Six points A,B,C,D,E and F are given. P, Q, R and S are the centroids of the triangles ABC, ABD, DEF and CEF respectively. The centroid of triangle ABC has the position vector 1/3 (a+b+c) and centroid of triangle ABD has the position vector 1/3 (a+b+d) given that position vector of D is d. (Position vectors of E and F are not given)

Show that P,Q,R,S are the vertices of a parallelogram.

Answer:

Hi,

  Given two centroids, we can obtain the position vectors R and S in a similar fashion (i.e. one-third of all vertices of the triangle):

   and .

  Now, what is left to be shown are the following:

   and

  which are straightforward. Try it!