$ABC$ is acute-angled. $D$ s a variable point on the side BC. $O_1$ is the circumcenter of $ABD$, $O_2$ is the circumcenter of $ACD$, and $O$ is the circumcenter of $AO_1O_2$. Find the locus of $O$.
1. **Identify the given elements and their properties:**
   - \( \triangle ABC \) is acute-angled.
   - \( D \) is a variable point on the side \( BC \).
   - \( O_1 \) is the circumcenter of \( \triangle ABD \).
   - \( O_2 \) is the circumcenter of \( \triangle ACD \).
   - \( O \) is the circumcenter of \( \triangle AO_1O_2 \).

2. **Establish the similarity relationships:**
   - Since \( \triangle AO_1O_2 \sim \triangle ABC \), we have:
     \[
     \frac{AO_1}{AB} = \frac{AO_2}{AC} = \frac{AO}{AX}
     \]
     where \( X \) is the circumcenter of \( \triangle ABC \).

3. **Use the properties of circumcenters:**
   - \( O_1 \) is equidistant from \( A \), \( B \), and \( D \).
   - \( O_2 \) is equidistant from \( A \), \( C \), and \( D \).
   - \( O \) is equidistant from \( A \), \( O_1 \), and \( O_2 \).

4. **Analyze the geometric configuration:**
   - Since \( \triangle AO_1B \sim \triangle AO_2C \sim \triangle AOX \), we have:
     \[
     \angle AO_1B = \angle AO_2C = \angle AOX
     \]
   - Given \( AO_1 = BO_1 \) and \( AO_2 = CO_2 \), it follows that \( AO = XO \).

5. **Determine the locus of \( O \):**
   - Since \( AO = XO \), \( O \) must lie on the perpendicular bisector of \( AX \).

6. **Verify the converse:**
   - For any point \( O \) on the perpendicular bisector of \( AX \), there exists a unique spiral similarity about \( A \) that maps \( X \) to \( B \) and \( O \) to \( O_1 \) such that \( AO_1 = BO_1 \).
   - This uniquely determines \( D \) on \( BC \), which in turn uniquely determines \( O_2 \) as the circumcenter of \( \triangle ACD \).

Conclusion:
The locus of \( O \) is the perpendicular bisector of \( AX \).

The final answer is the perpendicular bisector of \( \boxed{ AX } \).