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Semicircles are constructed on AB, AC, and BC.  A circle is tangent to all three semicircles.  Find the radius of the circle.

 

 Feb 13, 2024
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Let D be the center of the  circle and let its radius = r

 

Let E be the center of one of the small semi-circles

 

We can construct right triangle BDE  with EBD  = 90°

 

BE = 1

BD = 2 - r

ED =  1 + r

 

BE^2 + BD^2  = ED^2

 

1^1 + (2 -r)^2  = (1 + r)^2

 

1 + r^2 - 4r + 4  = r^2 + 2r + 1

 

6r =  4

 

r = 4/6  = 2/3

 

cool cool cool

 Feb 13, 2024

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