Let $a$ and $b$ be two positive real numbers, and $AB$ a segment of length $a$ on a plane. Let $C,D$ be two variable points on the plane such that $ABCD$ is a non-degenerate convex quadrilateral with $BC=CD=b$ and $DA=a$. It is easy to see that there is a circle tangent to all four sides of the quadrilateral $ABCD$.
Find the precise locus of the point $I$.
1. **Define the problem and given conditions:**
   - Let \( a \) and \( b \) be two positive real numbers.
   - \( AB \) is a segment of length \( a \) on a plane.
   - \( C \) and \( D \) are variable points such that \( ABCD \) is a non-degenerate convex quadrilateral.
   - \( BC = CD = b \) and \( DA = a \).
   - There exists a circle tangent to all four sides of the quadrilateral \( ABCD \).

2. **Identify the circle and its properties:**
   - Let \( \Gamma \) be the circle with center \( B \) and radius \( b \).
   - Consider a homothety (a transformation that scales distances from a fixed point) with center \( A \) and ratio \( \frac{a}{a+b} \). This homothety maps \( \Gamma \) to another circle \( \omega \).

3. **Determine the locus of point \( I \):**
   - The homothety maps the circle \( \Gamma \) to \( \omega \), and \( B \) lies on \( \omega \).
   - Let \( AX_1 \) and \( AX_2 \) be tangents to \( \omega \) at points \( X_1 \) and \( X_2 \), respectively.
   - The locus of the incenter \( I \) of the quadrilateral \( ABCD \) is the arc \( X_1BX_2 \) of \( \omega \), excluding the points \( X_1 \), \( X_2 \), and \( B \).

4. **Conclusion:**
   - The precise locus of the point \( I \) is the arc \( X_1BX_2 \) of the circle \( \omega \), excluding the points \( X_1 \), \( X_2 \), and \( B \).

The final answer is the arc \( \boxed{ X_1BX_2 } \) of the circle \( \omega \), excluding the points \( X_1 \), \( X_2 \), and \( B \).