A rhombus is inscribed in triangle $ABC$ such that one of its vertices is $A$ and two of its sides lie along $AB$ and $AC$. If $\overline{AC} = 6$ inches, $\overline{AB} = 12$ inches, and $\overline{BC} = 8$ inches, find the side of the rhombus in inches.

Let's think step by step and output the final answer within \boxed{}.
user