How to find the length of the triangle's edge in this geometry question containing excenter?

This is geogebra output of a question from my textbook.It can be not to scale by the way.

$B,D,F$ is linear. $E$ is the excenter. $EF$ is perpendicular to $[BF]$. In triangle $\triangle{ABD}$, $BE$ is the bisector of $\angle{ABD}$ and $DE$ is the bisector of $\angle{ADF}$. $|BD|=10$, $|AD|=8$ and $|DF|=5$. Find $|AB|.$

I tried to use bisector properties but i can't get the answer. I think we should use excenter. Any hint will be appreciated.

• Could you give the exact question, as written because it's not clear what you mean by "BE and DE is bisector"? Also, what have you tried to solve this problem? – B. Mehta Feb 28 '17 at 17:01
• DE is bisector of ADF? – Lozenges Feb 28 '17 at 17:56
• Sorry, i edited the question. – user373239 Feb 28 '17 at 17:59
• Maybe you should edit your picture instead. – Mick Feb 28 '17 at 18:05
• Which part i should edit? – user373239 Feb 28 '17 at 18:06

Given triangle $ABD$ and the excircle opposite angle $B$
Given $BF$ tangent to the excircle at $F$ and $BF=15$
$BA$ extended is tangent at $G$ and $BG=BF=15$. Therefore $AG=15-x$
$AD$ is tangent at $M$ with $AM=15-x$, $DM=5$, $AD=20-x=8$
$x=12$
Note: $x=AB$