Routh's theorem gives the area of a triangle determined by three cevians in a parent triangle, in terms of the "Ceva ratios" for those cevians.
After learning about Routh's theorem, I started thinking of possible generalizations. The question I came up with is
Can a similar formula be found for quadrilaterals with a similar construction?
That is, for convex quadrilateral $ABCD$, construct points $M$, $N$, $P$, $Q$ on sides $BC$, $CD$, $DA$, and $AB$, respectively, and take $S$, $T$, $U$, $V$ to be the intersections of lines $AM$ and $BN$, $BN$ and $CP$, etc.
Then, find an expression for the area of $\square STUV$ in terms of the ratios in which the points $M$, $N$, $P$, and $Q$ divide the sides of $\square ABCD$.
The method I thought of using to solve this problem was applying a transformation to $\square ABCD$, turning it into a square (so that the problem of finding the required area would be easy to solve), but I am not sure why such a transformation would preserve area ratios. In particular, I am interested in the case where points $M$, $N$, $P$, $Q$ lay in the middle of the sides of $\square ABCD$.
Thank you in advance!