smc

After drawing the diagram, we see that we actually have a lot of lengths to work with. Considering triangle ABD, we know values of $AB,BD(BD=BO+OD)$, but we want to find the value of $AD$. We can apply stewart's theorem now, letting $m=4,n=6,AD=X,AB=6$, and we have $10\cdot 6\cdot 4+8\cdot 8\cdot 10=x^2+36\cdot 6$, and we see that $x=\boxed{\sqrt{166}}$