Given a point D and a triangle ABC, suppose that line AD intersects line BC in point E and line BD intersects line CA in point F, and line CD intersects line AB in point G. Let the area of triangle ABC be S.

Problem 1. If BE/EC = 1/3, and CF/FA = 2, what is the ratio AG/GB? What are the barycentric coordinates of D? What is the area of triangle BCD (in terms of S)?

Problem 2. If BE/EC = 2/3, and CF/FA = -2, what is the ratio AG/GB? What are the barycentric coordinates of D? What is the area of triangle BCD (in terms of S)?

Problem 3. If AG/AB =1/4 and AF/AC = 4/5, what are the barycentric coordinates of D? What are the affine coordinates of D if A is taken to be the origin and B and C are the unit points on the axes?

Problem 4. Let point P be in the plane. Suppose there is a point B' on line AC so that PB' is parallel to line AB and a point C' on line AB so that PC' is parallel to line AC. Suppose we are given that the ratios BC'/BA = 3/8 and CB'/CA = 7/8. What are the barycentric coordinates of P? If line AP intersects line BC in point Q, what is the ratio BQ/QC? (Ratios in the figure below are NOT the same as in the problem.)

Problem 5. Given point A = (10,20,0) and B = (20,30,40), for what point P on line AB is the ratio AP/PB = 3/7?

Problem 6. Given a point A = (10,20,0) and B = (20,30,40), and C = (-20, 10, 30), for what point P are the barycentric coordinates of P with respect to ABC = (1/5,3/10,1/2)? If line AP intersects line BC at Q, what are the coordinates of point Q? Hint: Figure out Q as a center of mass.