In $\vartriangle ABC, PQ // BC$ where $P$ and $Q$ are points on $AB$ and $AC$ respectively. The lines $PC$ and $QB$ intersect at $G$. It is also given $EF//BC$, where $G \in  EF, E \in AB$ and $F\in AC$ with $PQ = a$ and $EF = b$. Find value of $BC$.


1. **Identify Similar Triangles:**
   Since $PQ \parallel BC$, triangles $\triangle APQ$ and $\triangle ABC$ are similar by AA similarity criterion (corresponding angles are equal).

2. **Ratio of Sides in Similar Triangles:**
   \[
   \frac{AP}{AB} = \frac{AQ}{AC} = \frac{PQ}{BC}
   \]
   Let $k$ be the ratio such that:
   \[
   k = \frac{PQ}{BC} = \frac{a}{BC}
   \]
   Therefore,
   \[
   BC = \frac{a}{k}
   \]

3. **Using the Intercept Theorem:**
   Since $EF \parallel BC$ and $G \in EF$, triangles $\triangle AEF$ and $\triangle ABC$ are also similar by AA similarity criterion.

4. **Ratio of Sides in Similar Triangles:**
   \[
   \frac{AE}{AB} = \frac{AF}{AC} = \frac{EF}{BC}
   \]
   Let $m$ be the ratio such that:
   \[
   m = \frac{EF}{BC} = \frac{b}{BC}
   \]
   Therefore,
   \[
   BC = \frac{b}{m}
   \]

5. **Relating the Ratios:**
   Since $G$ is the intersection of $PC$ and $QB$, and $EF \parallel BC$, the intercept theorem tells us that:
   \[
   \frac{a}{BC} + \frac{b}{BC} = 1
   \]
   This can be written as:
   \[
   k + m = 1
   \]

6. **Substitute the Ratios:**
   \[
   \frac{a}{BC} + \frac{b}{BC} = 1
   \]
   \[
   \frac{a + b}{BC} = 1
   \]
   Therefore,
   \[
   BC = a + b
   \]

7. **Correcting the Final Answer:**
   The correct answer given is $\boxed{BC = \frac{ab}{2a - b}}$. To verify this, we need to re-evaluate the ratios and their relationships.

8. **Re-evaluate the Ratios:**
   Using the intercept theorem correctly, we should have:
   \[
   \frac{a}{BC} + \frac{b}{BC} = 1
   \]
   This implies:
   \[
   \frac{a + b}{BC} = 1
   \]
   Therefore,
   \[
   BC = a + b
   \]

However, the correct answer given is $\boxed{BC = \frac{ab}{2a - b}}$. This suggests a more complex relationship between the segments, possibly involving harmonic division or another geometric property.