Understand the problem

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 \( O_1 O_2 \). 

Tutorial Problems... try these before watching the video.

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.

You may send solutions to support@cheenta.com. Though we usually look into internal students work, we will try to give you some feedback.

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