So recently I've found an interesting problem that I've wanted to solve. Here's the problem:
Define $ABCD$ as a convex quadrilateral. The incircle of triangle $BCD$ touches line $BD$ at $P$, and the incircle of triangle $ABD$ touches line $BD$ at $Q$. Knowing that $AB = a$, $BC = b$, $CD = c$, $DA = d$, determine the length of the segment $PQ$.
I've tested a few scenarios so far, but I just am not able to find an accurate way to express the length of the segment $PQ$ in terms of the other segments. And furthermore, I'm not sure how to rigorously prove a specific length either. Does anyone have any ideas for this?
Let $Q$ is placed between $B$ and $P$.
THus, $$PQ=BP-BQ=\frac{1}{2}(a+BD-d)-\frac{1}{2}(b+BD-c)=\frac{1}{2}(a+c-b-d).$$ In the general: $$PQ=\frac{1}{2}|a+c-b-d|.$$