1. Identify the key points and properties:
- Let $M$ be the midpoint of $AB$.
- Since $\triangle ABC$ is a right triangle with $\angle ACB = 90^\circ$, the centroid $G$ divides each median in the ratio $2:1$.
- The circumcircles $k_1$ and $k_2$ intersect $AB$ at points $P$ and $Q$ respectively.

2. **Determine the lengths involving the centroid $G$:**
- The centroid $G$ of $\triangle ABC$ is located at $\left( \frac{A + B + C}{3} \right)$.
- Since $M$ is the midpoint of $AB$, $CM = AM = BM = \frac{1}{2} AB$.

3. **Analyze the trapezoids $ACGP$ and $BCGQ$:**
- Since $G$ is the centroid, $GP \parallel AC$ and $GQ \parallel BC$.
- The lengths $AP$ and $BQ$ are each $\frac{2}{3} CM = \frac{1}{3} AB$.

4. **Calculate the distances involving $G$:**
- $GM = PM = QM = \frac{1}{6} AB$.

5. **Examine the rectangle $GPZQ$:**
- Since $GP \parallel AC$ and $GQ \parallel BC$, $GPZQ$ forms a rectangle.
- $M$ is the midpoint of $PQ$, so $CM$ passes through $Z$.

6. **Analyze the cyclic quadrilaterals $CGPX$ and $CGQY$:**
- From the cyclic quadrilateral $CGPX$, $\angle XCG = \angle XPG = 90^\circ$.
- Similarly, from $CGQY$, $\angle YCG = 90^\circ$.

7. **Determine the collinearity of points $X, C, Y$:**
- Since $\angle XCG = \angle YCG = 90^\circ$, points $X, C, Y$ are collinear.

8. **Analyze the right triangle $XYZ$:**
- $XYZ$ is a right triangle with height $ZC = ZM + CM = \frac{1}{6} AB + \frac{1}{2} AB = \frac{2}{3} AB$.

9. **Calculate $CX \cdot CY$:**
- Since $CX \cdot CY = ZC^2$, we have $ZC = \frac{2}{3} AB$.
- Therefore, $CX \cdot CY = \left( \frac{2}{3} AB \right)^2 = \frac{4}{9} AB^2$.

10. **Determine the ratio $\frac{CX \cdot CY}{AB^2}$:**
- $$\frac{CX \cdot CY}{AB^2} = \frac{\frac{4}{9} AB^2}{AB^2} = \frac{4}{9}$$

The final answer is $\boxed{\frac{4}{9}}$.