Area ratio divided in parallelogram

The quadrangle ABCD of the figure is a parallelogram. E, P and G are points on the sides AB, CD and DA, respectively and AE = EB and AG=GD and DF : FC = 4 : 1.

Moreover, the point H is an intersection of EF and BG.

The point I is an intersection of EF and CG.

Answer the following questions.

(1) The intersection of the straight line which extended side DA and the straight line which extended FE is set to P.

Moreover, the intersection of the straight line which extended the side BC and the straight line which extended EF is set to Q.

In this case, Find PA : AD and BC : CQ.

(2) Find GH : HB and GI : IC.

(3) Find the ratio of the area of the triangle AHG, the triangle GHI and the triangle GID.

Answer