A triangle $\vartriangle A_0A_1A_2$ in the plane has sidelengths $A_0A_1 = 7$,$A_1A_2 = 8$,$A_2A_0 = 9$. For $i \ge 0$, given $\vartriangle A_iA_{i+1}A_{i+2}$, let $A_{i+3}$ be the midpoint of $A_iA_{i+1}$ and let Gi be the centroid of $\vartriangle A_iA_{i+1}A_{i+2}$. Let point $G$ be the limit of the sequence of points $\{G_i\}^{\infty}_{i=0}$. If the distance between $G$ and $G_0$ can be written as $\frac{a\sqrt{b}}{c}$ , where $a, b, c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime, find $a^2 + b^2 + c^2$.
1. **Define the initial points and centroids:**
   - Given the triangle $\triangle A_0A_1A_2$ with side lengths $A_0A_1 = 7$, $A_1A_2 = 8$, and $A_2A_0 = 9$.
   - For $i \ge 0$, define $A_{i+3}$ as the midpoint of $A_iA_{i+1}$.
   - Let $G_i$ be the centroid of $\triangle A_iA_{i+1}A_{i+2}$.

2. **Calculate the coordinates of the midpoints:**
   - $A_3 = \frac{1}{2}(A_0 + A_1)$
   - $A_4 = \frac{1}{2}(A_1 + A_2)$
   - $A_5 = \frac{1}{4}(A_0 + A_1 + 2A_2)$
   - $A_6 = \frac{1}{4}(A_0 + 2A_1 + A_2)$

3. **Show similarity of triangles:**
   - Compute $A_4 - A_5$, $A_5 - A_6$, and $A_6 - A_4$ to show that $\triangle A_4A_5A_6$ is similar to $\triangle A_0A_1A_2$.
   - This similarity implies that the sequence of triangles converges to a point $G$.

4. **Determine the homothety factor:**
   - The scaling factor to get from one triangle to the next is $-\frac{1}{4}$.

5. **Find the centroid $G_0$ of $\triangle A_0A_1A_2$:**
   - The centroid $G_0$ divides the median $A_0A_4$ in the ratio $2:1$.

6. **Determine the position of $G$:**
   - The point $G$ divides the median $A_0A_4$ in the ratio $4:1$.

7. **Calculate the distance $|G_0G|$:**
   - The length of the median $A_0A_4$ is given by:
     \[
     A_0A_4 = \sqrt{\frac{1}{2}(7^2 + 9^2) - \frac{1}{4} \cdot 8^2} = \sqrt{\frac{1}{2}(49 + 81) - 16} = \sqrt{\frac{1}{2} \cdot 130 - 16} = \sqrt{65 - 16} = \sqrt{49} = 7
     \]
   - The distance $|G_0G|$ is:
     \[
     |G_0G| = \left| \frac{1}{3}A_0A_4 - \frac{1}{5}A_0A_4 \right| = \left| \frac{7}{3} - \frac{7}{5} \right| = \left| \frac{35}{15} - \frac{21}{15} \right| = \frac{14}{15}
     \]

8. **Identify the values of $a$, $b$, and $c$:**
   - Here, $a = 14$, $b = 1$, and $c = 15$.

9. **Calculate $a^2 + b^2 + c^2$:**
   - \[
   a^2 + b^2 + c^2 = 14^2 + 1^2 + 15^2 = 196 + 1 + 225 = 422
   \]

The final answer is $\boxed{422}$.