The distance between the incenter and circumcenter of a triangle, is calculated by Euler’s theorem in geometry: $$d=\sqrt{R(R-2r)}$$ which also implies $$R\ge2r$$
For a Bicentric quadrilateral, the distance between the incenter and circumcenter can be calculated by Fuss’s theorem or Carlitz’ identity.
Using Euler’s theorem in geometry, it can be approved that for $\triangle{ABC}$ $$1\le\cos{A}+\cos{B}+\cos{C}\le\dfrac{3}{2}$$ $$0\le\sin{\dfrac{A}{2}}\cdot\sin{\dfrac{B}{2}}\cdot\sin{\dfrac{C}{2}}\le\dfrac{1}{8}$$Because $$\cos{A}+\cos{B}+\cos{C}=1+\dfrac{r}{R}$$ $$4\cdot\sin{\dfrac{A}{2}}\cdot\sin{\dfrac{B}{2}}\cdot\sin{\dfrac{C}{2}}=\dfrac{r}{R}$$