3. A circle is inscribed in triangle $A B C$. Two points $E$ and $F$ are marked on the largest side of the triangle $A C$ such that $A E=A B$, and $C F=C B$. Segment $B E$ intersects the inscribed circle at points $P$ and $Q$, with $B P=1, P Q=8$. What is the length of segment $E F$?
Solution.

![](https://cdn.mathpix.com/cropped/2024_05_06_887daaf3f79b01753c00g-5.jpg?height=438&width=1009&top_left_y=124&top_left_x=267)

Let the sides of the triangle be $AB=c, AC=b, BC=a$. Then $EF=a+c-b$.

Let $G$ be the point of tangency of the inscribed circle with side $BC$. Then $BG^2 = BQ \cdot BP = 9$. Therefore, $BG = 3$.

Since the segments of tangents drawn from a point to a circle are equal, we have $2BG + 2b = a + b + c$.

Therefore, $a + c - b = 2BG = 6$.

Answer: 6.