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# 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.

[/et_pb_text][et_pb_text _builder_version="4.0.7" text_font_size="18px" custom_padding="20px|30px|20px|30px|false|false" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset1"]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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