Inscribed angle theorem

Statement

Let $A$, $B$ and $V$ be distinct points of a circle with the centre $O$ in the Euclidean plane $\mathbb{R}^2$. Let $\measuredangle$ be the oriented angle, with values in $\mathbb{R} / 2\pi\mathbb{Z}$. Then $$\measuredangle AOB = 2 \, \measuredangle AVB.$$

Remarks

For the unoriented angles, with values in $[0, \pi]$: if $V$ and $O$ are not on opposite sides of the line $AB$, then $\angle AVB = \frac{1}{2} \angle AOB$. If they are on opposite sides, then $\angle AVB = \pi - \frac{1}{2} \angle AOB$. Thus the inscribed angle does not change when $V$ moves on one arc from $A$ to $B$. Thales's theorem is the case where $AB$ is a diameter. The oldest known version is in Propositions 20 and 21 of Book III of Euclid's Elements.

Explanations

L1 human

Fix two points $A$ and $B$ on a circle. The angle at the centre that looks at the arc $AB$ is twice the angle at any point of the circle on the other arc. The proof starts with the case where the line $VO$ passes through $B$. Then the triangle $VOA$ is isosceles, and its angle sum gives the result.