Let \(ABC\) be a triangle. \(D\) and \(E\) lie on \(AB\) such that \(AD = AC, BE = BC\) and the points \(D, A, B, E\) are collinear in that order. The bisectors of angle \(A\) and \(B\) intersect \(BC, AC\) at \(P\) and \(Q\) respectively, and the circumcircle of \(ABC\) at \(M\) and \(N\) respectively. Let \(O_1\) be the circumcenter of \(BME\) and \(O_2\) be the circumcenter of \(AND\). \(AO_1\) and \(BO_2\) intersect at \(X\). Prove that \(CX\) is perpendicular to \(PQ\).
Source : Serbia \(2008\).