Let a semicircle $AB$ be given and let $X$ be an inner point of the arc. Consider a point $Y$ on ray $XA$ such that $XY=XB$. Find the locus of all points $Y$ when $X$ moves on the arc $AB$ (excluding the endpoints).
1. **Identify the given elements and setup:**
   - Let the semicircle have diameter \(AB\) and center \(O\).
   - Let \(X\) be a point on the arc \(AB\) (excluding the endpoints \(A\) and \(B\)).
   - Let \(Y\) be a point on the ray \(XA\) such that \(XY = XB\).

2. **Determine the properties of the angle bisectors:**
   - The angle bisectors of \(\angle AXB\) concur at the midpoint \(M\) of the arc \(AB\). This is because \(M\) is the center of the circle that subtends the arc \(AB\) and thus lies on the perpendicular bisector of \(AB\).

3. **Reflect \(B\) across \(XM\):**
   - Since \(Y\) is the reflection of \(B\) across \(XM\), we need to find the geometric properties of this reflection.
   - The reflection of a point across a line is equidistant from the line. Therefore, \(Y\) will lie on the circle centered at \(M\) with radius equal to the distance from \(M\) to \(B\).

4. **Determine the locus of \(Y\):**
   - As \(X\) moves along the arc \(AB\), the line \(XM\) changes its orientation, but \(M\) remains fixed.
   - The point \(Y\) will always be the reflection of \(B\) across \(XM\), meaning \(Y\) will trace out a path that is symmetric with respect to \(M\).

5. **Conclude the locus:**
   - The locus of all points \(Y\) is the minor arc \(AB\) of the circle centered at \(M\) and passing through \(A\) and \(B\). This is because \(Y\) is always the reflection of \(B\) across the line \(XM\), and as \(X\) moves, \(Y\) traces out the same arc.

\(\blacksquare\)

The final answer is the minor arc \( \boxed{ AB } \) of the circle centered at \(M\) and passing through \(A\) and \(B\).