Let $ABC$ be a triangle with sides $AB = 4$ and $AC = 6$. Point $H$ is the projection of vertex $B$ to the bisector of angle $A$. Find $MH$, where $M$ is the midpoint of $BC$.
1. **Identify the given elements and their properties:**
   - Triangle \(ABC\) with \(AB = 4\) and \(AC = 6\).
   - Point \(H\) is the projection of vertex \(B\) onto the angle bisector of \(\angle A\).
   - Point \(M\) is the midpoint of \(BC\).

2. **Extend \(BH\) to meet \(AC\) at point \(D\):**
   - Since \(H\) is the projection of \(B\) onto the angle bisector of \(\angle A\), \(H\) lies on the angle bisector.
   - By the Angle Bisector Theorem, \(AD\) is the angle bisector of \(\angle A\), and it divides \(BC\) into segments proportional to the other two sides of the triangle:
     \[
     \frac{BD}{DC} = \frac{AB}{AC} = \frac{4}{6} = \frac{2}{3}
     \]

3. **Determine the lengths of \(BD\) and \(DC\):**
   - Let \(BD = 2x\) and \(DC = 3x\). Since \(BC = BD + DC\), we have:
     \[
     BC = 2x + 3x = 5x
     \]
   - The total length of \(BC\) is not given, but we can use the midpoint property later.

4. **Use the midpoint property:**
   - Since \(M\) is the midpoint of \(BC\), \(BM = MC\).
   - Therefore, \(BM = \frac{BC}{2} = \frac{5x}{2}\).

5. **Analyze the projection \(H\):**
   - Since \(H\) is the projection of \(B\) onto the angle bisector \(AD\), \(BH \perp AD\).
   - By the properties of projections and midpoints, \(MH\) is half the length of \(DC\) because \(H\) is the midpoint of \(BD\) and \(D\) is the midpoint of \(AC\).

6. **Calculate \(MH\):**
   - Since \(D\) is the midpoint of \(AC\), \(AD = DC = 3\) (half of \(AC\)).
   - Therefore, \(MH = \frac{DC}{2} = \frac{3}{2} = 1.5\).

The final answer is \(\boxed{1.5}\).