Let \( \Gamma_1 \) and \( \Gamma_2 \) be two circles with unequal radii, with centers \( O_1 \) and \( O_2 \) respectively, in the plane intersecting in two distinct points A and B. Assume that the center of each of the circles \( \Gamma_1 \) and \( \Gamma_2 \) are outside each other. The tangent to \( \Gamma_ 1 \) at B intersects \( \Gamma_2 \) again at C, different from B; the tangent to \( \Gamma_2 \) at B intersects \( \Gamma_1 \) again in D different from B. The bisectors of \( \angle DAB \) and \( \angle CAB \) meet \( \Gamma_1 \) and \( \Gamma_2 \) again in X and Y, respectively. different from A. Let P and Q be the circumcenters of the triangles ACD and XAY, respectively. Prove that PQ is perpendicular bisector of the line segment \( O_1 O_2 \).
1. Suppose \( P O_1 Q O_2 \) be a kite (that is \( PO_1 = PO_2 \) and \( QO_1 1 = QO_2 \). Show that PQ is perpendicular bisector of the other diagonal $ O_1 O_2 $.$.
2. Show that for any two circles intersecting each other at two distinct points, the common chord is bisected perpendicularly by the line joining the center.
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