Parellelograms and polynomials together, Really?

Geometry Level 5

The lengths of the sides of a triangle \(ABC\) with centroid at \(G\) are the roots of the

cubic polynomial:

\[k(x)=x^3-25x^2+200x-500\]

If \(P, Q\) and \(R\) are the points in the plane of the triangle such that

\(PAGB, QBGC\) and \(RAGC\) are parallelograms. Then find the value of:

\(PG^2+QG^2+RG^2\)

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