External angle between two tangents = half the arc difference

Depends on: Tangent is perpendicular to the radius at the point of tangency, Central ∠, arc, chord are pairwise equivalent.

Statement

Let $\odot O$ be the circle with centre $O$, and let $P$ be a point outside $\odot O$. From $P$ draw two tangents to $\odot O$, touching at $A$ and $B$. The two tangents cut $\odot O$ into two arcs: the arc closer to $P$ is the near arc $\widehat{AB}_{\text{near}}$ (minor arc), and the arc farther from $P$ is the far arc $\widehat{AB}_{\text{far}}$ (major arc). Then the angle $\angle APB$ between the two tangents equals half the arc difference:

Here arc "measure" is taken to be the corresponding central angle; near arc + far arc = $360^{\circ}$.