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Let ABC be a triangle with circumcircle \omega and incentre I. A line \ell intersects the lines AI, BI, and CI at points D, E, and F, respectively, distinct from the points A, B, C, and I. The perpendicular bisectors x, y, and z of the segments AD, BE, and CF, respectively determine a triangle \Theta. Show that the circumcircle of the triangle \Theta is tangent to \Omega.

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