I am looking for the proof of the following claim:
Consider a family of bicentric quadrilaterals with the same inradius length. Denote by $P$ and $Q$ the midpoints of the diagonals, and by $I$ the incenter. Then, $|PI| cdot |QI|$ has the same value for all quadrilaterals in the family.
The GeoGebra applet that demonstrates this claim can be found here.