I can use the rule but I have no idea where the 1/2 comes from, so I forget which one gets halved.
Accepted answer
The clean case: let the inscribed angle be at point `A`, with one side passing through the center `O`, hitting the circle at `B`, and the other side at `C`.
Because `OA` and `OC` are both **radii**, triangle `AOC` is isosceles, so the angles at `A` and `C` are equal — call each `x`.
The central angle `∠BOC` is an **exterior angle** of triangle `AOC`, so it equals the sum of the two non-adjacent interior angles: `x + x = 2x`.
The inscribed angle is `x`, the central angle is `2x`. Half.
Cases where the center isn't on a side split into (or subtract) two of these, giving the same result. The halving belongs to the **inscribed** angle because the isosceles triangle splits the central angle in two — that's the direction to remember.