The central angle equals twice the inscribed angle on the same arc

Dependencies: Base angles of an isoceles triangle are equal (base angles of an isosceles triangle are equal), Exterior angle equals the sum of two non-adjacent interior angles (an exterior angle equals the sum of the two non-adjacent interior angles).

Statement

Let $A$, $B$, $P$ be three points on $\odot O$, with $P$ not on the segment of line $AB$ between $A$ and $B$ (i.e. $P$ lies on the "far-side" arc subtended by chord $AB$ and $O$). $\angle APB$ is called the inscribed angle subtending arc $AB$, and $\angle AOB$ the central angle subtending the same arc. Then

In other words, the inscribed angle is exactly half the central angle on the same arc.