# Where Do They Intersect?

Geometry Level 3

Let $$\Gamma_1$$ and $$\Gamma_2$$ be two concentric circles, and let the radius of $$\Gamma_1$$ be greater than that of $$\Gamma_2.$$ Consider two non-parallel lines $$\ell_1$$ and $$\ell_2$$ passing through $$\Gamma_2.$$ One of the intersections of $$\ell_1$$ and $$\Gamma_2$$ is $$P;$$ one of the intersections of $$\ell_2$$ and $$\Gamma_2$$ is $$Q.$$ Let $$R$$ be one of the intersections of $$\ell_1$$ and $$\Gamma_1,$$ and let $$S$$ be the second intersection point of $$\Gamma_1$$ and the circumcircle of $$\triangle PQR.$$ Suppose $$S$$ lies on $$\ell_2.$$ Lines $$\ell_1$$ and $$\ell_2$$ meet at $$X.$$ Suppose $$XP= XQ.$$ Where does $$X$$ lie?

I did not include a picture because I'm afraid that even an inaccurate diagram might give the answer away.

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