Telltale signs

Geometry Level 4

Consider an acute triangle \(ABC\) with \(X\) as the foot of the perpendicular from \(A\) to \(BC\). Let the circle \(\Omega\) with \(AX\) as the diameter intersect \(AB\) and \(AC\) at \(D\) and \(E\) respectively. Let \(BE\) and \(CD\) intersect circle \(\Omega \) at \(E'\) and \(D'\) respectively. If the measures of the angles \(E'AX, EE'D', D'AX \) and \(EBC\) are \(a,b,c\) and \(d\) respectively, and \( \dfrac{\sin (a) \sin(b)}{\sin(c) \sin(d)} = \dfrac xy\) for coprime positive integers \(x\) and \(y\), calculate \(x+y\).

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