Weird Ratio

Geometry Level 3

In \(\triangle{ABC}\), Let \(D\) and \(E\) be the trisection points of \(BC\) with \(D\) between \(B\) and \(E\). Let \(F\) be the midpoint of \(AC\), and let \(G\) be the midpoint of \(AB\). Let \(H\) be the intersection of \(EG\) and \(DF\). If the ratio \(EH:HG\) equals \(a:b\) where \(a,b\) are relatively prime positive integers. Find \(a+b\).

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