In the diagram, $TQ$ and $PR$ are the diameters of the circle with centre $O$.
The tangent at $T$ meets $RP$ produced at $U$. Prove that

1. Identify equal side lengths from the properties of a circle

Since $TQ$ and $PR$ are diameters intersecting at the centre $O$,
$OT = OQ$ (radii of circle)
$OR = OP$ (radii of circle)

2. Identify equal angles

$\angle TOR = \angle QOP$ (vertically opposite angles)

3. Prove congruence and conclude the result

By SAS congruence test, $\triangle OTR \cong \triangle OQP$.
Therefore, $TR = PQ$.