In the figure, \(\triangle ABC\) is equilateral with area \(S\). \(M\) is the mid-point of \(BC\), and \(P\) is a point on \(AM\) extended such that \(MP = BM\). If the semi-circle on \(AP\) intersects \(CB\) extended at \(Q\), and the area of a square with \(MQ\) as a side is \(T\), which of the following is true?