Inspired by Calvin Lin: Any part 1

Geometry Level 4

Suppose \(AD, BE, CF\) are the medians of a triangle \(ABC\), with side \(BC = 10\), intersecting at \(G\).

In the quadrilateral \(AEGF\), suppose \( AE\times AF = 3\times GE\times GF \),
evaluate \(\dfrac{AB\times BE+ AC\times CF}{AD}\).

Inspired by Calvin's recent median potpourri!
Do solve the next part.

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