Euclidean cirle question Let $c_1$ be a circle with center $O$. Let angle $ABC$ be an inscribed angle of the circle $c_1$. 
i) If $O$ and $B$ are on the same side of the line $AC$, what is the relationship between $\angle ABC$ and $ \angle AOC$?
ii) If $O$ and $B$ are on opposite side of the line $AC$, what is the relationship between $\angle ABC$ and $\angle AOC$
I guess $\angle ABC=(1/2) \angle AOC$ but I don't know how to explain


Here $O$ and $B$ are on the same side of of the line $AC$, you can figure out the other part.
$\angle AOB=180-2y$ and $\angle COB=180-2x$ 
$\angle AOB+ \angle COB=360-2(x+y) \implies \angle AOC=2(x+y) \implies 2 \angle ABC$.
This is known widely as Inscribed Angle theorem as RobJohn said in his comment.:)
