# 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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