In the diagram, P, Q, R and S are points on a circle. PR and QS meet at T.

1. Identify one pair of equal angles at the intersection of the chords.

Since $PR$ and $QS$ intersect at $T$,

$\angle PTS = \angle QTR$

because they are vertically opposite angles.

2. Identify a second pair of equal angles using the circle theorem.

Since $T$ lies on $QS$, $\angle PST = \angle PSQ$.

Since $T$ lies on $PR$, $\angle QRT = \angle QRP$.

Both $\angle PSQ$ and $\angle QRP$ are angles subtended by the same chord $PQ$ at the circumference.

Therefore,

$\angle PST = \angle QRT$

because angles in the same segment are equal.

3. Conclude the triangles are similar by AA similarity.

Since two pairs of corresponding angles are equal,

$\triangle PTS \sim \triangle QTR$

by AA similarity.

1. Use the similarity from part (a) to form a ratio of corresponding sides.

From part (a), $\triangle PTS \sim \triangle QTR$.

Corresponding sides are proportional, so

$\frac{PT}{QT}=\frac{ST}{RT}.$

2. Cross-multiply to obtain the required product equality.

Cross-multiplying,

$PT \times RT = QT \times ST.$

Therefore,

$QT \times ST = PT \times RT.$