Triangle-Ception I

Geometry Level 5

Equilateral triangle \(ABC\) has side lengths of \(12\). A cevian \(AD\) is drawn from \(A\) to a point \(D\) on \(BC\). A segment \(DE\) is then drawn from point \(D\) to a point \(E\) on \(AC\). Lastly, a segment \(EF\) is drawn from point \(E\) to a point \(F\) on \(AD\).

Given that the areas of triangles \(ABD,CDE,DEF,\) and \(AEF\) are all equal, the length of \(EF^{2}\) is equal to \(\frac{p}{q}\) where \(p\) and \(q\) are co-prime positive integers. Find the value of \(p+q\).

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