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Question: The mid-points of AB and DC of parallelogram ABCD are E and F respectively. AF and EC intersect the diagonals BD at points P and Q respectively. BP:PD = How much?

In the problem described, points E and F are the mid-points of AB and DC respectively. So, the line segment EF is the mid-line in triangle ABC and triangle ADC. In triangles, the mid-line is parallel to the base and its length is equal to a half of the base. Since E and F are midpoints, AF and EC each divides parallelogram ABCD into two equal area triangles. Also, the intersection point of diagonals in a parallelogram divides the diagonals into two equal parts. Therefore, point P and Q should coincide and should be the midpoint of BD. Therefore, BP:PD = 1:1.

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