In a quadrilateral ABCD, the diagonals intersect at O. Let the area of the triangle ABD be p. If AO : OC = m : n, then what is the area of the triangle BCD?

Triangles ABD and BCD share the same base BD. Their areas are in the ratio of the perpendiculars from A and C to BD. Since AO:OC = m:n along the diagonal AC, and A, O, C are collinear with the foot relationship, the perpendicular distances from A and C to BD are in ratio m:n. So area(ABD)/area(BCD) = m/n, giving area(BCD) = (n/m)·p = np/m.