Question 9
Let $I$ be the incenter of a triangle $ABC$ satisfying $AB \gt AC$, and let $D, E$ be points on the sides $AB, AC$, respectively, dividing the sides into two segments with $1:8$ ratio. If the triangle $DIE$ becomes the regular triangle with side length $1$, what is the length of $AB$? Here by $XY$ we denote the length of the line segment $XY$ as well.