Mathematics

The key insight is the inscribed angle theorem: all angles inscribed in a circle that subtend the same arc are equal. Since both $C$ and $D$ lie on the same side of chord $AB$, they both "see" the same arc $AB$ from the circumference. This means $\angle ADB$ must equal $\angle ACB$, regardless of where exactly $D$ sits on that arc. The reason is that an inscribed angle depends only on the arc it subtends, not on the specific position of the vertex along the arc.

The key insight is the inscribed angle theorem: all angles inscribed in a circle that subtend the same arc are equal. Since both $C$ and $D$ lie on the same side of chord $AB$, they both "see" the same arc $AB$ from the circumference. This means $\angle ADB$ must equal $\angle ACB$, regardless of where exactly $D$ sits on that arc. The reason is that an inscribed angle depends only on the arc it subtends, not on the specific position of the vertex along the arc.