A circle with center $O$ and a point $A$ in this circle are given. Let $P_{B}$ is the intersection point of $[AB]$ and the internal bisector of $\angle AOB$ where $B$ is a point on the circle such that $B$ doesn't lie on the line $OA$, Find the locus of $P_{B}$ as $B$ varies.
1. **Given:**
   - A circle with center \( O \) and a point \( A \) inside the circle.
   - A point \( B \) on the circle such that \( B \) does not lie on the line \( OA \).
   - \( P_B \) is the intersection point of the segment \([AB]\) and the internal bisector of \(\angle AOB\).

2. **Objective:**
   - Find the locus of \( P_B \) as \( B \) varies on the circle.

3. **Analysis:**
   - Let \( \angle AOB = \theta \).
   - The internal bisector of \(\angle AOB\) will intersect \([AB]\) at \( P_B \).

4. **Using the Angle Bisector Theorem:**
   - The Angle Bisector Theorem states that the internal bisector of an angle in a triangle divides the opposite side into segments that are proportional to the adjacent sides.
   - Therefore, \(\frac{AP_B}{P_BB} = \frac{OA}{OB}\).

5. **Coordinate Geometry Approach:**
   - Assume \( O \) is at the origin \((0,0)\).
   - Let \( A \) be at \((a,0)\) and \( B \) be at \((r\cos\theta, r\sin\theta)\), where \( r \) is the radius of the circle.

6. **Finding \( P_B \):**
   - The coordinates of \( P_B \) can be found using the section formula:
     \[
     P_B = \left( \frac{a \cdot OB + 0 \cdot OA}{OA + OB}, \frac{0 \cdot OB + r\sin\theta \cdot OA}{OA + OB} \right)
     \]
     Since \( OA = a \) and \( OB = r \):
     \[
     P_B = \left( \frac{a \cdot r\cos\theta}{a + r}, \frac{r\sin\theta \cdot a}{a + r} \right)
     \]

7. **Simplifying the Coordinates:**
   - Let \( k = \frac{a}{a + r} \):
     \[
     P_B = \left( k \cdot r\cos\theta, k \cdot r\sin\theta \right)
     \]
   - Notice that \( k \) is a constant since \( a \) and \( r \) are constants.

8. **Locus of \( P_B \):**
   - As \( B \) varies on the circle, \(\theta\) varies from \(0\) to \(2\pi\).
   - The coordinates of \( P_B \) describe a circle with radius \( kr \) centered at the origin.

9. **Conclusion:**
   - The locus of \( P_B \) is a circle with radius \( \frac{a \cdot r}{a + r} \) centered at the origin.

The final answer is \(\boxed{\text{A circle with radius } \frac{a \cdot r}{a + r} \text{ centered at the origin.}}\)