1. Identify the given elements and their properties:
- $O$ is the circumcenter of the isosceles triangle $ABC$ with $AB = AC$.
- $P$ is a point on the segment $AO$.
- $Q$ is the symmetric of $P$ with respect to the midpoint of $AB$.
- $OQ$ intersects $AB$ at $K$.
- The circle passing through $A, K,$ and $O$ intersects $AC$ at $L$.

2. Establish the symmetry and congruence relationships:
- Since $Q$ is the symmetric of $P$ with respect to the midpoint of $AB$, $Q$ lies on the perpendicular bisector of $AB$.
- $O$ is the circumcenter, so $OA = OB = OC$.
- The circle passing through $A, K,$ and $O$ implies that $K$ and $L$ are concyclic with $A$ and $O$.

3. Use the properties of the circumcircle and isosceles triangle:
- Since $AB = AC$, $\angle BAC = \angle BCA$.
- $O$ being the circumcenter means $\angle BOA = \angle COA = 2\angle BAC$.

4. Analyze the intersection points and angles:
- Since $OQ$ intersects $AB$ at $K$, and $K$ lies on the circle passing through $A, K,$ and $O$, we have $\angle KAO = \angle KLO$.
- By the property of the circle, $\angle KAO = \angle OCA$.

5. Prove the congruence of triangles:
- $\Delta KOA \cong \Delta LOC$ by the properties of the circle and the isosceles triangle.
- This implies $OK = OL$ and $\angle KAO = \angle OCA$.

6. Use the symmetry to establish angle relationships:
- Since $Q$ is symmetric to $P$ with respect to the midpoint of $AB$, $\angle QBK = \angle PAL$.
- $\Delta BQK \cong \Delta APL$ by the symmetry and congruence properties.

7. Conclude the angle equality:
- From the congruence $\Delta BQK \cong \Delta APL$, we have $\angle ALP = \angle BQK$.
- Since $\angle BQK = \angle CLO$ by the properties of the circumcircle and symmetry, we conclude $\angle ALP = \angle CLO$.

$$\boxed{\angle ALP = \angle CLO}$$