Circles and straight lines

Geometry Level 5

\(Six~ points~(x_{i},y_{i}),~i=1,2,......6\) are taken on the circle \(x^2+y^2=4\) such that \(\displaystyle \sum _{i=1}^{6}x_{i}=8~and~\displaystyle \sum_{i=1}^{6}y_{i}=4\). The line segment joining \( orthocentre \) of the triangle made by any three points and the \(centroid\) of the triangle made by other three points passes through a fixed point \((h,k)\), then find \(h+k\)

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