perplexus.info :: Geometry : Incircle in a semicircle

Incircle in a semicircle (Posted on 2023-11-30)

A semicircle with radius 1 is centered at O, and has diameter AB. Point P lies on the semicircle so that angle ABP=60 degrees. Compute the radius of the circle tangent to diameter AB, segment BP, and minor arc AP.

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

| Comment 2 of 3 |

Call the point of tangency of the circle with AB by C.

Call the center of the circle by D

If CD=r, the radius of circle then CB=r*sqrt(3)

OC=1-r*sqrt(3)

By pythagorus on triangle OCD

OD=sqrt(1-2r*sqrt(3)+4r^2)

Extending ray OD will intersect the semicircle at the point of tangency so we get the equation

sqrt(1-2r*sqrt(3)+4r^2) + r = 1

Which solves nicely

r=2(sqrt(3)-1)/3 or about 0.488

Posted by Jer on 2023-11-30 13:59:53

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