6. Let $C_1$ and $C_2$ be dijoint circles with centers $O_1$ and $O_2$. A common exterior tangent touches $C_1$ and $C_2$ at $A$ and $B$, respectively. Line segments $O_1O_2$ cuts $C_1$ and $C_2$ at point $C$ and $D$,respectively. Prove that the points $A,B,C$ and $D$ are concyclic.
7.In $\Delta ABC$, the line through $C$ parallel to the bisector of $\hat{B}$ cuts the bisector of $\hat{A}$ at $D$. The line through $C$ parallel to the bisector of $\hat{A}$ cuts the bisector of $\hat{B}$ at $E$.If $DE$ is parallel to $AB$, prove that $ABC$ is an isosceles triangle.
8.Circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$,respectively. intersect at points $A $and $B$ so that the center of each circle lies outside the other circle.Line $O_1A $intersects circle $k_2$ again at point $P_2$ and line $O_2A$ intersects circle $k_1$ again at $P_1$. Prove that the points $O_1,O_2,P_1,P_2, B$ are concyclic.
9.Let $O$ be the center of the circle touching the side $AC$ of triangle $ABC$ and the extensions of the sides $BA$ and $BC$. $D$ is the center of the circle passing through the point $A,B$ and $O$. Prove that the point $A,B,C$ and $D$ are concyclic.
10.Let triangle $ABC$ have orthocenter $H$, and let $P$ be a point on its circumcircle, distinct from $A,B,C$ . Let $E$ be the foot of the altitude $BH$, let $PAQB$ and $PARC$ be parallelograms, and let $AQ$ meet $HR$ in $X$.Prove that $EX$ is parallel to $AP$
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