How to find the center point and radius of a circle given two sides and a single point 
I have a wooden shelf (blue) with a support (red). The support is 17 cm long and 5 cm from the right edge of the shelf. I want to cut a part of the board that is in the lower right corner so that the support does not stick out underneath the board but the board is "as round as possible" on that corner. The green circle is there just to cut that one corner and is not a real physical object. How can i find the center point (as x y coordinates) and radius of the green circle?
I am sawing an actual shelf and it also needs to be done on the left side of the shelf.
EDIT: for clarity, the only relevant part of the circle is the bottom right. The circle is "fixed" to the bottom and the right side of the shelf. Also the top of the red line seems to be on the circle, but this is irrelevant, only the bottom of the red line should be on the circle.
 A: 
The question can be restated as:

find the circle,  inscribed in the
lower right corner,  which passes through the point $F$ (the lower end
of the support.

Using similarity, the center of the sought circle can be found by construction as follows.

*

*Draw the line $BG$ that bisects the lower right corner
(the center of the circle must be located on it).


*Choose any point $O_1$ on that line and draw circle $\mathcal{C_1}$ with the radius $r_1$
such that it touches the bottom and the right side.


*Find the point $F_1$ at the intersection of the line $FB$ and the circle $\mathcal{C_1}$


*find the scaling factor
\begin{align}
k&=\frac{|FB|}{|F_1B|}
.
\end{align}


*The radius of the sought circle is $r=k\cdot r_1$.

For the given dimensions, the value of the sought radius can be found as
\begin{align}
r&=18+\sqrt{130}
\approx 29.40 \mathrm{cm}
.
\end{align}
Also, the center $O$ of the sought circle can be found
at the intersection of the line $BG$
and the line through the point $F$
parallel to the line $F_1O_1$.

Edit

In general, for given shelf length $|AB|=|CD|=l$,
width $|AD|=|BC|=w$,
length of the support $|EF|=v\le w$
and offset  $|CE|=u$, the radius of the circle can be found as
\begin{align}
r&=u+w-v+\sqrt{2u(w-v)}
.
\end{align}
