Proof for a nice conjecture related to circles I have made a conjecture but am unable to prove it.

If $P$ and $Q$ are any points such that they are both at an equal given distance from a given point $C$, subtending a fixed angle at another given point $O$ (wherein $ OC<PC=QC $). Then the maximum and minimum distance $PQ$ occurs when $P$ and $Q$ lie on equally inclined rays to the line perpendicular to $OC$ through $O$.

Clearly, the minimum case would be in the smaller segment and maximum in the larger segment. Also note  that if $P$ and $Q$ subtend an angle $2x$ at $O$, then each of the rays will be inclined at $90°-x$ to the line perpendicular to $OC$.

Consider the circle $a$ through $OPQ$ (violet in figure below). If this circle is not tangent to circle $c$ of center $C$ passing through $O$ (dashed), then on $c$ there are points both inside $a$ (as $O'$ in the figure) and outside $a$ (as $O''$). But then
$$
\angle PO''Q<\angle POQ<\angle PO'Q
$$
and from that it follows that it's possible to construct an angle of vertex $O'$ equal to $\angle POQ$ but subtending an arc on $a$ smaller than $PQ$, and it's possible to construct an angle of vertex $O''$ equal to $\angle POQ$ but subtending an arc on $a$ larger than $PQ$.
But the same angles can be constructed from vertex $O$ (just rotate them about $C$), hence if $a$ is not tangent to $c$ then point $O$ cannot subtend a minimum or maximum arc.
It follows that point $O$ subtends a minimum or maximum arc only if circle $POQ$ is tangent to circle $c$, which is equivalent to the thesis to be proved.

