Placing a pole with maximal radius subject to constraints

[GuyWhoOnceToldYou]

TL;DR Summary

Looking to place a pole with maximal radius, constrained by objects surrounding it

I have a problem that I imagine does not have a closed-form solution and requires the use of some kind of optimization solver. I am not an engineer myself, so forgive me if the question seems stupid.

The problem is as follows: I have a circle bound in a square, and an arm going from the center of the circle to the corner of the square (so, 45 degrees), and I need to place a circle with the largest radius possible in the area bound between the outside of the big circle, the inside of the square, and below the arm (the area in green in the image)

I can describe these constraints mathematically: I want to maximize some r subject to the constraints: let 2*w be the total width of the arm, and r_circle be the radius of the black circle, then the constraints are:

(r+r_circle)*(cos(alpha),sin(alpha)) is at least distance w+r from the line y=x (I can directly calculate the closest point on the line as a function of r and alpha to simplify this one, I just haven't done that yet).
(r+r_circle)*(cos(alpha),sin(alpha)) is at least distance r from the line from the line x=r_circle (the right edge of the square) (again, I can calculate the point as a function of r, alpha directly)
alpha is between 0 and 45 degrees


So what I'm looking for is either (A) a better way to solve this, or (B) a tool that can solve this.

Thanks in advance!

 

[Baluncore]

Where the big circle has radius R;
The small circle will have a radius; r = R * 0.123308641756286

R = 1.
Small circle centred at x = 1 - r ; y = x - r * Sqrt(2)
Distance between centres d = Sqrt( x*x + y*y ) ;
Find r, for d - r = 1 ;

 

[GuyWhoOnceToldYou]

Small circle centred at x = 1 - r ; y = x - r * Sqrt(2)

Of course! I can assume the resulting circle will touch both x=1 and the arm at 45 degrees, that didn't occur to me. Thank you! I will have to adjust to account for the width of the arm, but I think that would be easy.

 

[Baluncore]

Welcome to PF.

0.123308641756286 = 1 / ( 3 + √2 + 2*√( 2 + √2 ) )
0.123308641756286 = 5 - 3*√2 - 2*√( 10 - 7*√2 )