Τρίτη, 10 Οκτωβρίου 2017

2008 JBMO Shortlist G10

Let $\Gamma$ be a circle of center $O$, and $\delta$.  be a line in the plane of $\Gamma$, not intersecting it. Denote by $A$ the foot of the perpendicular from $O$ onto $\delta$., and let $M$ be a (variable) point on $\Gamma$. Denote by $\gamma$ the circle of diameter $AM$ , by $X$ the (other than M ) intersection point of $\gamma$  and $\Gamma$, and by $Y$ the (other than $A$) intersection point of $\gamma$  and $\delta$. Prove that the line $XY$  passes through a fixed point.

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