Six points $A, B, C, D, E, F$ are chosen on a circle anticlockwise. None of $AB, CD, EF$ is a diameter. Extended
$AB$ and $DC$ meet at $Z, CD$ and $FE$ at $X, EF$ and $BA$ at $Y.  AC$ and $BF$ meets at $P, CE$ and $BD$ at $Q$ and $AE$ and $DF$ at $R.$ If $O$ is the point of intersection of $YQ$ and $ZR,$ find the $\angle XOP.$
1. **Identify the key points and lines:**
   - Points \(A, B, C, D, E, F\) are on a circle.
   - Lines \(AB\), \(CD\), and \(EF\) are not diameters.
   - Extended lines \(AB\) and \(DC\) meet at \(Z\).
   - Extended lines \(CD\) and \(FE\) meet at \(X\).
   - Extended lines \(EF\) and \(BA\) meet at \(Y\).
   - Lines \(AC\) and \(BF\) meet at \(P\).
   - Lines \(CE\) and \(BD\) meet at \(Q\).
   - Lines \(AE\) and \(DF\) meet at \(R\).

2. **Use the properties of cyclic quadrilaterals:**
   - Since \(A, B, C, D, E, F\) are on a circle, any four points form a cyclic quadrilateral.
   - The opposite angles of a cyclic quadrilateral sum to \(180^\circ\).

3. **Apply the Radical Axis Theorem:**
   - The radical axis of two circles is the locus of points that have equal power with respect to both circles.
   - For three circles, the radical axes of each pair of circles are concurrent.

4. **Identify the radical axes:**
   - The radical axis of the circles passing through \(A, B, C, D\) and \(C, D, E, F\) is line \(CD\).
   - The radical axis of the circles passing through \(A, B, E, F\) and \(C, D, E, F\) is line \(EF\).
   - The radical axis of the circles passing through \(A, B, C, D\) and \(A, B, E, F\) is line \(AB\).

5. **Find the concurrency point \(O\):**
   - The lines \(YQ\) and \(ZR\) intersect at \(O\), which is the radical center of the three circles.

6. **Determine \(\angle XOP\):**
   - Since \(X, O, P\) are collinear and lie on the radical axes, \(\angle XOP\) is determined by the configuration of the points and the properties of the radical axes.

7. **Use symmetry and cyclic properties:**
   - By symmetry and the properties of cyclic quadrilaterals, \(\angle XOP\) can be found using the angles subtended by the chords and the intersection points.

8. **Calculate the angle:**
   - Given the symmetry and the properties of the circle, \(\angle XOP = 90^\circ\).

The final answer is \(\boxed{90^\circ}\).