{
  "aliases": [
    "central-angle-theorem"
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  "body": "\n## Statement\n\nLet $A$, $B$ and $V$ be distinct points of a [[circle|060W5CBZP6]] with the centre $O$ in the Euclidean plane $\\mathbb{R}^2$. Let $\\measuredangle$ be the [[oriented angle|03YN70NK0Z]], with values in $\\mathbb{R} / 2\\pi\\mathbb{Z}$.\nThen\n$$\\measuredangle AOB = 2 \\, \\measuredangle AVB.$$\n\n## Remarks\n\nFor the unoriented [[angles|0GJAJC6TNZ]], 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 $\\angle AVB = \\pi - \\frac{1}{2} \\angle AOB$.\nThus the inscribed angle does not change when $V$ moves on one arc from $A$ to $B$.\n[[Thales's theorem|0GSDHYMA1T]] is the case where $AB$ is a diameter.\nThe oldest known version is in Propositions 20 and 21 of Book III of Euclid's Elements.\n\n## Explanations\n\n### L1 human\n\nFix 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.\n",
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    "aliases": [
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    "name": "Inscribed angle theorem",
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  "name": "Inscribed angle theorem",
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