Tangent–chord angle equals the inscribed angle on the same side

Depends on: Tangent is perpendicular to the radius at the point of tangency (tangent ⊥ radius), Central angle is twice the inscribed angle on the same arc (Central angle is twice the inscribed angle on the same arc, including the Thales right-angle special case), Inscribed angles on same arc are equal; angle in a semicircle is right (inscribed angles on the same arc are equal); the appendix "chain proof" additionally uses Base angles of an isoceles triangle are equal.

Statement

Let $\ell$ be the tangent to $\odot O$ at the point $T$, and let $TA$ be a chord of $\odot O$. The angle between $\ell$ and $TA$ at $T$ is called the tangent–chord angle, denoted $\alpha$. The chord $TA$ cuts $\odot O$ into two arcs; pick any point $P$ on the arc that lies on the same side as $\alpha$ (i.e. on the same side of $TA$ as the opening of $\alpha$). Then the inscribed angle $\angle TPA = \gamma$ satisfies

In other words, the tangent–chord angle equals the inscribed angle on the same-side arc — and equals half of that same-side arc.