As is the case with triangles, I am sure that there is an entire ocean of not-so-well-known theorems about Euclidean quadrilaterals. In particular, I am interested in the following problem and suspect the existence of a theorem that elegantly trivializes it (the black points are the mid-points of the respective sides). What would that theorem be?
Call the midpoint of $BC$ $M$ and midpoint of $AD$ $N$, and $P$ the intersection of $AM$ and $BN$, and $Q$ the intersection of $MD$ and $CN$. Let $t$ and $y$ be the areas of triangles $PMN$ and $QMN$ respectively. Then: $t+81 = y+72$ and $t+49 = y+x$. Thus: $32 = 72 - x \implies x = 40$.