It's not hard to prove why this should be true. I'll use the diagram from the post above:

-angle QPO is x;
-OP and OQ are both radii, so they're equal. That makes OPQ isosceles, and angle PQO must also be x;
-the angles in OPQ must add to 180, so angle POQ must be 180-2x;
-POS is a straight line, so POQ + QOS must be 180. That makes QOS equal to 2x, or exactly double angle QPO.

If you connect QS, you can continue to fill in angles here to discover that the angle PQS will always be 90 degrees, another useful fact to know.