Maximum Ratio of Areas

Geometry Level 5

Let \(T\) be an acute-angled triangle, and \(R\),\(S\) be rectangles inscribed in it as shown. If \(A(x)\) denotes the area of the figure \(x\), the maximum value of \(\displaystyle \dfrac{A(R)+A(S)}{A(T)}\) can be expressed as \(\dfrac{P}{Q}\), where \(P,Q\) are coprime positive integers. Find the maximum possible area of a triangle with \(P\), \(Q\) as two of it's sides.

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