The quadrilateral $ABCD$ with perpendicular diagonals is inscribed in the circle with center $O$, the points $M,N$ are the midpoints of $[BC]$ and $[CD]$ respectively. Find the ratio of areas of the figures $OMCN$ and $ABCD$
1. **Identify the given information and setup the problem:**
   - Quadrilateral \(ABCD\) is inscribed in a circle with center \(O\).
   - The diagonals \(AC\) and \(BD\) are perpendicular.
   - Points \(M\) and \(N\) are the midpoints of segments \([BC]\) and \([CD]\) respectively.

2. **Reflect point \(C\) with respect to \(O\):**
   - Let \(C'\) be the reflection of \(C\) with respect to \(O\). Since \(O\) is the center of the circle, \(C'\) will also lie on the circle and \(OC = OC'\).

3. **Analyze the perpendicularity and similarity:**
   - Since \(AC \perp BD\) and \(AC' \perp AC\), it follows that \(AC' \perp BD\).
   - The triangles \(\triangle ABD\) and \(\triangle C'BD\) are congruent because they share the same base \(BD\) and have equal heights (since \(C'\) is the reflection of \(C\)).

4. **Calculate the area relationships:**
   - The area of \(\triangle ABD\) is equal to the area of \(\triangle C'BD\).
   - Since \(\triangle OMN\) is similar to \(\triangle C'BD\) by the reflection property and the fact that \(M\) and \(N\) are midpoints, the area of \(\triangle OMN\) is \(\frac{1}{4}\) of the area of \(\triangle C'BD\).

5. **Determine the area of quadrilateral \(OMCN\):**
   - The area of quadrilateral \(OMCN\) is composed of two triangles, \(\triangle OMC\) and \(\triangle OCN\), each of which is half of \(\triangle OMN\).
   - Therefore, the area of \(OMCN\) is twice the area of \(\triangle OMN\), which is \(\frac{1}{2}\) of the area of \(\triangle C'BD\).

6. **Relate the areas of \(OMCN\) and \(ABCD\):**
   - Since \(\triangle ABD\) and \(\triangle C'BD\) are congruent, the area of \(ABCD\) is twice the area of \(\triangle ABD\).
   - Given that \(\triangle ABD = \triangle C'BD\), the area of \(ABCD\) is four times the area of \(\triangle OMN\).

7. **Conclude the ratio of the areas:**
   - The area of \(OMCN\) is \(\frac{1}{4}\) of the area of \(ABCD\).

The final answer is \(\boxed{\frac{1}{4}}\).