Triangle-Ception III

Geometry Level 4

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

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

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

Find m+n+pm+n+p.


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