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

Vxrtate Feb 13, 2024

#1**+1 **

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

CPhill Feb 13, 2024