In a triangle $ABC$, $\angle BAC=90^{\circ }$. Let $D$ be the point on segment $BC$ such that $AB+BD=AC+CD$. Suppose $BD:DC=2:1$. If $\frac{AC}{AB}=\frac{m+\sqrt{p}}{n}$, where $m,n$ are relatively prime positive integers and $p$ is a prime number, determine the value of $m+n+p$.

Step 1: Let $AB=y$. Since $BD:DC=2:1$, we can let $CD=x$ and $BD=2x$. The total length of the hypotenuse is $BC=BD+CD=3x$.

Step 2: The given relation is $AB+BD=AC+CD$. Substituting our variables:

Step 3: Since $△ABC$ is a right-angled triangle with $\angle BAC=90^{\circ }$, by the Pythagorean theorem:

Step 4: Expand and simplify the equation:

Step 5: Divide the entire equation by $x^2$ and let $u=y/x$:

Solving this quadratic equation for $u>0$:

Step 6: We want to find the ratio $\frac{AC}{AB}$:

Step 7: Calculate $\frac{1}{u}$:

Step 8: Add $1$ to find the ratio:

Comparing this to $\frac{m+\sqrt{p}}{n}$, we get $m=9$, $n=8$, and $p=17$.

Step 9: Since $9$ and $8$ are relatively prime and $17$ is prime, these values satisfy the conditions. Finally, we calculate the sum:

Thus, the value of $m+n+p$ is $34$.