Let $O$ be a point on $\Gamma$ such that $O$ is the midpoint of the arc $AB$ not containing $C$. We have, \[\angle AOB = 180 - \angle ACB = 180 - (180 - 2\angle ADB) = 2\angle ADB.\]

TheDarkPrince wrote: Let $O$ be a point on $\Gamma$ such that $O$ is the midpoint of the arc $AB$ not containing $C$. We have, \[\angle AOB = 180 - \angle ACB = 180 - (180 - 2\angle ADB) = 2\angle ADB.\] Why do you consider $O$ as the mid-point only? Even if you consider $O$ to be any point (obviously not coinciding with either $A$ or $B$) on the arc ${AB}$ not containing $C$, your logic remains true..