Three circles are constructed on a triangle, with the medians of the triangle forming the diameters of the circles. It is known that each pair of circles intersects. Let $C_{1}$ be the point of intersection, further from the vertex C, of the circles constructed from the medians $AM_{1}$ and $BM_{2}$. Points $A_{1}$ and $B_{1}$ are defined similarly. Prove that the lines $AA_{1}$, $BB_{1}$ and $CC_{1}$ intersect at the same point.

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