# Inscribed circles under partial circles

Calculus Level 5

We have a semi circle of radius 1 with diameter $$AB$$. A line $$AC$$ is drawn making an angle $$\theta$$ with $$AB$$ and intersecting the semicircle at $$C$$.

Now a circle is drawn such that it touches line $$AB, AC$$ and the arc $$BC$$ inside the semicircle.

Let the radius of the circle be $$r$$, if $$r$$ can be written as $$r= f(\theta)$$, find $$\displaystyle 6\int _{ 0 }^{ 1/2 }{ f(2\tan ^{ -1 }{ \theta )d\theta } }$$.

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