Question

What is the relationship between angle BAC and angle BOC?

Answer 1

In this problem, an angle like angle BAC where the vertices like on the circle itself is called the inscribed angle.

While angle BOC, where O is the center of the circle, is called the central angle.

Using  Proposition III.20 from Euclid's Elements, this is called the Inscribed Angle Theorem  wherein:

∠BOC = 2∠BAC

or ∠BOC / 2 = ∠BAC