In the diagram, W is a point on the circle with centre O - NSC Mathematics - Question 10 - 2017 - Paper 2

To prove that MN || TS, we observe that the angles subtended by the same arc are equal. Therefore, we can establish that:

$\angle OWT = \angle OMV$

Since these angles are equal, by the corresponding angles criterion, we conclude that MN || TS.

To prove that TMNS is a cyclic quadrilateral, we need to show that the opposite angles are supplementary. We can ascertain:

$\angle MTS + \angle MNS = 180^\circ$

This is derived from the fact that angles subtended by the same chord are equal, which fulfills the requirement for TMNS being a cyclic quadrilateral.

To prove the relationship between OS, MN, ON, and WS:

1. We first establish that OV = VN, as given in the problem statement.
2. Then, since we have the angles in triangles AOV and AOW equal, we can use similarity: $\frac{OS}{ON} = \frac{WS}{VN}$.
3. Applying the property of similar triangles, we can multiply both sides by ON and substitute to derive the equation:
   $OS \cdot MN = 2ON \cdot WS$.