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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