Two-secant / two-chord angle formulas

Dependencies: Central angle is twice the inscribed angle on the same arc (inscribed angle = half the central angle, equivalently = half the arc), Exterior angle equals the sum of two non-adjacent interior angles (a triangle's exterior angle equals the sum of the two non-adjacent interior angles).

Statement

Let $\odot O$ be a circle and $P$ a point in the plane.

Case (external): $P$ is outside $\odot O$, and from $P$ we draw two secants — one meeting the circle at $A$ (near) and $B$ (far), the other at $C$ (near) and $D$ (far). Then the angle subtended at $P$ between the two secants equals half the difference between the far arc $\widehat{BD}$ and the near arc $\widehat{AC}$:

Case (internal): $P$ is inside $\odot O$, and through $P$ we draw two chords $AB$ and $CD$ that meet at $P$. Then the angle subtended at $P$ equals half the sum of the two opposite arcs:

Here "opposite arcs" means the two arcs cut off on each side of $\angle APC$ — one arc $\widehat{AC}$ (contained in $\angle APC$) on one side, and another arc $\widehat{BD}$ (contained in $\angle BPD$, the vertical angle) on the other.