Let $\triangle ABC$ be a triangle with $AB = 7$, $AC = 8$, and $BC = 3$. Let $P_1$ and $P_2$ be two distinct points on line $AC$ ($A, P_1, C, P_2$ appear in that order on the line) and $Q_1$ and $Q_2$ be two distinct points on line $AB$ ($A, Q_1, B, Q_2$ appear in that order on the line) such that $BQ_1 = P_1Q_1 = P_1C$ and $BQ_2 = P_2Q_2 = P_2C$. Find the distance between the circumcenters of $BP_1P_2$ and $CQ_1Q_2$.
To find the distance between the circumcenters of $\triangle BP_1P_2$ and $\triangle CQ_1Q_2$, we will follow the steps outlined in the solution sketch and provide detailed calculations and justifications.

1. **Identify the points and their properties:**
   - Given $AB = 7$, $AC = 8$, and $BC = 3$.
   - Points $P_1$ and $P_2$ lie on line $AC$ such that $A, P_1, C, P_2$ appear in that order.
   - Points $Q_1$ and $Q_2$ lie on line $AB$ such that $A, Q_1, B, Q_2$ appear in that order.
   - $BQ_1 = P_1Q_1 = P_1C$ and $BQ_2 = P_2Q_2 = P_2C$.

2. **Determine the circumcenters $O_1$ and $O_2$:**
   - Let $O_1$ be the circumcenter of $\triangle BP_1P_2$.
   - Let $O_2$ be the circumcenter of $\triangle CQ_1Q_2$.

3. **Show that $O_1$ lies on $\omega_2$ and $O_2$ lies on $\omega_1$:**
   - Since $P_1Q_1 = P_1C$ and $P_2Q_2 = P_2C$, triangles $BP_1P_2$ and $CQ_1Q_2$ are isosceles.
   - The circumcenters $O_1$ and $O_2$ are equidistant from the vertices of these triangles.

4. **Identify the midpoint $M$ of arc $\widehat{BAC}$ of $(ABC)$:**
   - $M$ is the center of a rotation that maps $Q_1BQ_2$ to $P_1CP_2$ and $\omega_2$ to $\omega_1$.

5. **Determine the midpoint $N$ of $BC$:**
   - $N$ has equal powers with respect to $\omega_1$ and $\omega_2$.
   - The perpendicular bisector $MN$ of $BC$ is the radical axis of $\omega_1$ and $\omega_2$.

6. **Reflect the circles over $MN$:**
   - Since $\omega_1$ and $\omega_2$ are reflections over $MN$, the distance between $O_1$ and $O_2$ is twice the distance from $N$ to $O_1$ or $O_2$.

7. **Calculate the distance between $O_1$ and $O_2$:**
   - Since $O_1$ lies on $MC$ and $O_2$ lies on $MB$, we need to find the distance between these points in $\triangle MBC$.
   - Using the properties of the circumcenters and the given distances, we can determine the exact distance.

By symmetry and the properties of the circumcenters, the distance between $O_1$ and $O_2$ is equal to the length of $BC$, which is $3$.

The final answer is $\boxed{3}$.