Meta-Medians

Geometry Level 5

In a triangle \(ABC\) the points \(D,E, F\) are located on sides \(AB,BC,\) and\(CA\) respectives so that \(\frac{AF}{AB}= \frac{BD}{BC}=\frac{CE}{CA}=\frac{1}{3}\). Find the value of the ratio \[\frac{AD^2+BE^2+CF^2}{AB^2+BC^2+CA^2}\]

If the result is in the form \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, report \(a+b\).

Inspiration: Triangle Centroid by Worranat Pakornrat

Note: Drawing is not to scale.

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