I have a following problem:
I have been trying to solve it for the last 2 days, but haven't got any luck so far.
I was trying to create two systems of equations for two points where the inner circle intersects the two quarter circles. I have also taken into account that points where the inscribed circle intersects the quarter circles have the same y coordinate and that they are symmetrical about the line $x = \frac12$.
What I have got so far is (assuming that $x_1$ is the x coordinate of the left intersection point and $x_2$ - of the right intersection point, $r$ is the radius of the inscribed circle and its center is at $(\frac12, r)$ ):
$$x_1^2+y^2=1$$ $$(x_1-\frac12)^2+(y-r)^2=r^2$$ $$(x_1-1)^2+y^2=1$$ $$(x_2-\frac12)^2+(y-r)^2=r^2$$
Unfortunatelly, no matter how I tried to simplify, rearrange and solve these equations I always end up with something like $0 = 0$, so I can't get the exact result.
Any help would be appreciated. Thanks in advance.