Converse of central angle is double angle subtended at circumference

Question

I have an exercise to explain why the following theorem does not have a converse:

Theorem.

The angle subtended at the centre of the circle by a chord is double the angle subtended at the circumference.

But why is the following not the converse?

Suppose $k$ is a circle centered at $O$ with $P,Q\in k$. Then if $\angle POQ=2\angle PRQ$ for some point $R$, then $R\in k$.

Edit: This appears to be true (see attempted proof in comments). So why does the book say there is no converse?

Reference: Crossing the Bridge by Gerry Leversha, p.64

Answer 1

Here $R'$ us the mirror image of $R$ with respect to $PQ$. Even though the relation between angles hold, it does not have to lie on $k$, aka $\odot O$. This proves that there is no converse of the theorem.