Let's suppose the points C, D, E can be anywhere in the interior and BF<10.
If C=A, D=B, E=A we get
|BC|+|CD|+|DE|+|EF| = |AB|+|AB|+|AB|+|FA| = 10 + 10 + 10 + 9 = 39
As the points are in the interior, this won't ever exactly equal 39 so the answer is 38
If BF > 10 then we can actually get a larger answer (thanks Roberto!) so we conclude this isn't the best question ever asked. If it has to be an integer it can't be 19 so it must be 18. Then our maximum is
18 + 18 + 10 + 9 - 1 = 3(18) = 54
