$D$ is midpoint of $AC$ for $\triangle ABC$. Bisectors of $\angle ACB,\angle ABD$ are perpendicular. Find max value for $\angle BAC$

[i](S. Berlov)[/i]
1. **Given**: \( D \) is the midpoint of \( AC \) in \( \triangle ABC \). The bisectors of \( \angle ACB \) and \( \angle ABD \) are perpendicular. We need to find the maximum value of \( \angle BAC \).

2. **Setup**: Let \( \angle BAC = \alpha \), \( \angle ABC = \beta \), and \( \angle ACB = \gamma \). Since \( D \) is the midpoint of \( AC \), we have \( AD = DC \).

3. **Angle Bisectors**: The angle bisector of \( \angle ACB \) intersects \( AB \) at some point \( E \). Since the bisectors of \( \angle ACB \) and \( \angle ABD \) are perpendicular, we use the angle bisector theorem and properties of perpendicular bisectors.

4. **Angle Relationships**: Since \( D \) is the midpoint of \( AC \), we have:
   \[
   \angle ADB = \angle BDC = \frac{\gamma}{2}
   \]
   The angle bisector of \( \angle ABD \) intersects \( AC \) at \( E \). By the angle bisector theorem:
   \[
   \frac{AE}{ED} = \frac{AB}{BD}
   \]
   Since \( D \) is the midpoint of \( AC \), \( AD = DC = \frac{b}{2} \).

5. **Perpendicular Bisectors**: Given that the bisectors of \( \angle ACB \) and \( \angle ABD \) are perpendicular, we have:
   \[
   \angle ACB + \angle ABD = 90^\circ
   \]
   Since \( \angle ABD = \frac{\alpha + \beta}{2} \), we get:
   \[
   \gamma + \frac{\alpha + \beta}{2} = 90^\circ
   \]

6. **Using Law of Cosines**: Let \( AC = b \), \( AB = c \), and \( BC = a \). Using the Law of Cosines in \( \triangle ABC \):
   \[
   \cos(\alpha) = \frac{b^2 + c^2 - a^2}{2bc}
   \]

7. **Maximizing \( \alpha \)**: To find the maximum value of \( \alpha \), we need to maximize \( \cos(\alpha) \). Given the perpendicularity condition, we have:
   \[
   \cos(\alpha) = \frac{a^2 + c^2}{2\sqrt{2}ac} \geq \frac{1}{\sqrt{2}}
   \]
   This implies:
   \[
   \cos(\alpha) \leq \frac{1}{\sqrt{2}}
   \]
   Therefore:
   \[
   \alpha \leq 45^\circ
   \]

8. **Equality Condition**: Equality holds when \( a = c = \frac{b}{\sqrt{2}} \). This configuration satisfies the given conditions and maximizes \( \alpha \).

Conclusion:
\[
\boxed{\angle BAC \leq 45^\circ}
\]