Geometry

$(1+r)^2=(\overline{AB}-r)^2+1^2\\(1+r)^2=(2-r)^2+1\\ 1+2r+r^2=4-4r+r^2+1\\ 6r=4\\\color{blue}r=\frac{2}{3}$

$\color{blue}The\ radius\ of\ the\ circle\ is\ \frac{2}{3}.$

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