Semi-semicircle

Geometry Level 4

Take a semicircle of radius \(R\) and one semicircle \(k\) times bigger and make them tangent like in the picture above. Then construct a \({{90}^{\circ }}\) arc through the midpoints of these 2 semicircles. If the shaded area is \(A\), calculate the ratio \(\dfrac{A}{kR^2}\).

Hint: Notice that all three centers are collinear.

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