Problem
Given an arbitrary complex quadrilateral ABCD, draw lines BE and DF from vertices B and D to the midpoints E and F of the opposite side. This divides the quadrilateral into two triangles, shaded blue, and a quadrilateral (shaded pink).
The question asked which is greater: the proportion of the quadrilateral that is shaded blue or the proportion that is shaded pink.
Solution
If a question asks 'Which is greater, X or Y?' the answer is often neither, they are the same, and so it is here. To see this, we add some extra lines.
Consider triangles ABE (blue) and EBD (pink). They share the same height GB and because E is the midpoint of AD they have the equal base lengths. So they must have equal areas.
A similar argument applies to triangles BDF (pink) and FDC (blue) and shows their areas are also equal.
In both cases the blue and pink areas are equal and therefore the overall proportions shaded pink and blue respectively must be the same.