The incircle of a triangle $ABC$ touches the side $AB$ and $AC$ at respectively at $X$ and $Y$. Let $K$ be the midpoint of the arc $\widehat{AB}$ on the circumcircle of $ABC$. Assume that $XY$ bisects the segment $AK$. What are the possible measures of angle $BAC$?
1. **Define Points and Given Information:**
   - Let $ABC$ be a triangle with incircle touching $AB$ at $X$ and $AC$ at $Y$.
   - Let $K$ be the midpoint of the arc $\widehat{AB}$ on the circumcircle of $\triangle ABC$.
   - Given that $XY$ bisects the segment $AK$.

2. **Introduce Additional Points and Properties:**
   - Let $L$ be the midpoint of the arc $\widehat{AC}$ on the circumcircle of $\triangle ABC$.
   - Let $I$ be the incenter of $\triangle ABC$.

3. **Apply Fact 5:**
   - By Fact 5, we know that $IK = AK$ and $IL = AL$.
   - This implies that $K$ and $L$ are equidistant from $I$ and $A$ respectively.

4. **Perpendicular Bisector and Parallel Lines:**
   - Since $IK = AK$ and $IL = AL$, the line $KL$ is the perpendicular bisector of $AI$.
   - Also, $XY$ is perpendicular to $AI$ because $XY$ is the angle bisector of $\angle BAC$ and it bisects $AK$.

5. **Parallel Lines:**
   - Since both $KL$ and $XY$ are perpendicular to $AI$, it follows that $KL \parallel XY$.

6. **Intersection and Ratio:**
   - Let $AI \cap XY = T$.
   - Given that $XY$ bisects $AK$, it is not difficult to see that $3AT = IT$.

7. **Angle Calculation:**
   - The condition $3AT = IT$ implies a specific geometric configuration.
   - This configuration is consistent with $\angle BAC = 120^\circ$.

Thus, the measure of angle $BAC$ is $120^\circ$.

The final answer is $\boxed{120^\circ}$