Inscribed circles under partial circles

Calculus Level 5

We have a semi circle of radius 1 with diameter ABAB. A line ACAC is drawn making an angle θ\theta with ABAB and intersecting the semicircle at CC.

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

Let the radius of the circle be rr, if rr can be written as r=f(θ) r= f(\theta) , find 601/2f(2tan1θ)dθ \displaystyle 6\int _{ 0 }^{ 1/2 }{ f(2\tan ^{ -1 }{ \theta )d\theta } } .

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