Does the line connecting the mid-points of two opposite sides of a quadrilateral divide it equally?

Question

Suppose ABCD is a general quadrilateral. P and Q are the mid points of AB and CD respectively. Now, will PQ line divide ABCD in two equal quadrilateral? That is, will the area of APQD and BPQC be equal?

Answer 1

It won't. If the quadrilateral is not convex, it won't even divide the area into two parts at all. But for the convex case consider what happens when $D$ tends to $A$: you get a triangle $ABC$, and $PQ$ will link the midpoints of $AB$ and $AC$, so it will divide the area by a ratio $1:3$. That is the limit situation; it is inconceivable that during the deformation to the limit situation the ratio would be $1:1$ throughout.

Answer 2

No. Just take your usual isosceles trapezoid and divide it in half. You can literally flip the top half into the bottom and you will still have a couple of triangles left over :)