1. The problem statement, all variables and given/known data In triangle ABC, the point P bisects the side BC, the point Q is the point on side AB for which |AQ| = 2|QB|, and the point R is the point of intersection between AP and CQ. Designate vectors a=AB and b=AC. Describe vector AR in terms of vectors a and b. 2. Relevant equations 3. The attempt at a solution I drew a triangle representation for the problem: http://img338.imageshack.us/img338/2571/triangulus.jpg [Broken] I can't find a way to express either of vectors AR, PR, QR or CR in terms of AB, AC, BC, CQ or AP no matter how much I've tried vector addition and subtraction alone. I realized that triangles ABP and APC are equal in area, and triangle QBC has half the area of triangle AQC. Consequently, I tried to use the fact that the cross product of vectors CQ and CB is equal to the area of triangle AQC, but I've consistently failed to find a good relationship between AR and other vectors. Attempts to describe the areas of the smaller triangles, or the quadrilateral QBPR in terms of other parts haven't gotten me anywhere either. Any hints would be appreciated.