1. In a quadrilatecral OABC, D is the midpoint of BC and E is the point on AD such that AE : ED = 2:1. Given that OA = a, OB = b and OC = c, express OD and OE in terms of a, b and c. 2. Prove that the line joining the midpoint of a median to a vertex of the triangle trisects the side opposite the vertex considered. 3. In a parallogram ABCD, P divides AB in the ratio 2:5 and Q divides DC in the ratio 3:2. If AC and PQ intersect at R, find the ratios AR : RC and PR: RQ. 4. D and E are any two points on the sides AB and AC respectively of the triangle ABC. DG drawn parallel to BE meets AC in G and EF drawn parallel to CD meets AB in F. Prove that FG is parallel to BC. 5. The points D, E and F divide the sides BC, CA and AB of a triangle in the ratios 1:4, 3 :2 and 3 :7 respectively. Show that there exists a point G such that the sum of the vectors AD, BE, CF is parallel to CG. What ratio does G divide AB? 6. ABCD is a plane quadrilateral and E is any point on AD. EF is drawn parallel to DB to meet AB in F, and EG is drawn parallel to DC to meet AC in G. Prove that FG is parallel to Bc. 7. G and G are the centroids of the two triangles ABC and A'BC". Show that AA' + BB' + CC = 3GG. 8. ABCD is a trapezium such that DC = 3AB. If L and M are the midpoints of AD and BC respectively, Show that LM = 2AB. 9. The points D, E and F divide the sides BC, CA and AB of a triangle in the ratio 1:4, 3:2 and 3 :7 respectively. Show that there exists a point G such that the sum of the vectors AD, BE, CF is parallel to CG. What is the ratio in which G divides AB?