# Triangle-Ception III

Geometry Level 4

In the figure above, $$\triangle ABF \sim \triangle BCE$$, with $$\angle FAB = \angle EBC$$ and $$\angle AFB = \angle BEC=109^{\circ}$$.

Additionally, points $$A,B,C,D$$ are collinear and points $$D,E,F$$ are collinear.

Given that $$AC=8$$ and $$BD=6$$, the length of $$AD$$ can be expressed in the form $$m+n\sqrt{p}$$ where $$m,n,p$$ are positive integers, and $$p$$ is not divisible by the square of any prime.

Find $$m+n+p$$.

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