Dynamic Geometry: P98

Geometry Level pending

The diagram shows a black semicircle. The cyan and green semicircles are tangent to each other and internally tangent to the black semicircle. They are growing and shrinking freely so that the sum of their radius is always equal to the black semicircle's radius. We draw a red vertical segment using their tangency point. At last we inscribed two yellow circles so they are tangent to the red line, to the black semicircle and to one of the two bottom semicircles. Using the tagency points, we draw a purple triangle and a blue triangle. When the ratio of the purple triangle's area to the area of the blue triangle is equal to 23281565\dfrac{2328}{1565}, the ratio of the cyan semicircle's radius to the radius of the green radius can be expressed as pq\dfrac{p}{q}, where pp and qq are coprime positive integers. Find qp\sqrt{q}-\sqrt{p}.

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