RMO 2015

Geometry Level 4

Let \(ABC\) be a triangle with circumcircle 􀀀 and incentre \(I\). Let the internal angle bisectors of angles \(A, B\), and \(C\) meet 􀀀the circumcircle in \(X, Y\), and \(Z\) respectively. Let \(YZ\) intersect \(AX\) in \(P\) and \(AC\) in \(Q\), and let \(BY\) intersect \(AC\) in \( R\). Suppose the quadrilateral \(PIRQ\) is a kite; that is, \(IP = IR\) and \(QP = QR\). The radius of the circumcircle is 2 and the area of triangle \(ABC\) is expressed as \(d^{3/2} \), find the value of \(\sqrt d\).

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