1. Identify the given elements and their relationships:
- $AB$ is a diameter of the circle with center $O$.
- $CD$ is a chord perpendicular to $AB$.
- $AE$ is a chord intersecting $CO$ at $M$.
- $DE$ and $BC$ intersect at $N$.

2. Establish the harmonic quadrilateral:
- Since $AB$ is a diameter, $\angle AOB = 180^\circ$.
- $\square ACBD$ is a harmonic quadrilateral because $AB$ is a diameter and $CD$ is perpendicular to $AB$.

3. Use the properties of harmonic division:
- In a harmonic quadrilateral, the cross-ratio $(A, B; C, D)$ is $-1$.
- This implies that $(B, A; D, C)$ is also $-1$.

4. Consider the intersection points and their properties:
- Let $AE$ cut $BC$ at $F$.
- Since $\square ACBD$ is harmonic, the cross-ratio $(B, A; D, C)$ is preserved under projection.

5. Apply the cross-ratio preservation:
- Projecting from $E$, we have $(B, A; D, C) = (B, F; N, C)$.
- Projecting from $M$, we have $(B, F; N, C) = (B, A; X, O)$, where $X$ is the point at infinity because $AB$ is parallel to $MN$.

6. Use the properties of parallel lines:
- Since $AB$ is parallel to $MN$, the cross-ratio $(B, A; X, O)$ simplifies to $(B, A; \infty, O)$.
- This implies that $AO = BO$ and $AB$ is parallel to $MN$.

7. Establish the required ratio:
- Since $AB$ is parallel to $MN$, the triangles $\triangle CMO$ and $\triangle CBN$ are similar by AA similarity criterion.
- Therefore, $\frac{CM}{CO} = \frac{CN}{CB}$.

$$\boxed{\frac{CM}{CO} = \frac{CN}{CB}}$$