Semicircles are constructed on AB, AC, and BC. A circle is tangent to all three semicircles. Find the radius of the circle.
Find the radius of the circle.
\((1+r)^2=(\dfrac{\overline{AC}}{2}-r)^2+1^2\\ (1+r)^2=(\dfrac{4}{2}-r)^2+1^2\\ 1+2r+r^2=4-4r+r^2+1\\ 6r=4\\ \color{blue}r=\dfrac{2}{3}\)
The radius of the circle is \(\color{blue}\frac{2}{3}\ .\)
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