{
  "aliases": [
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  "body": "\n## Statement\n\nLet $A$, $B$, $C$ and $D$ be [[concyclic points|049DCJZ448]] of the Euclidean plane $\\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Suppose that the chords $AC$ and $BD$ meet at a point $S$ that is inside both segments.\nThen\n$$d(A, S)\\, d(S, C) = d(B, S)\\, d(S, D).$$\n\n## Remarks\n\nBoth products are equal to $r^2 - d(S, O)^2$, where $O$ is the centre and $r$ the radius. This is minus the [[power|02E9VT1RVQ]] of $S$.\nA proof: the triangles $ASD$ and $BSC$ are similar, by the inscribed angle theorem.\nThe theorem is Proposition 35 of Book III of Euclid's Elements.\n\n## Explanations\n\n### L1 human\n\nDraw two chords of a circle that cross each other. Each chord is cut into two pieces. The product of the two pieces is the same for both chords.\n",
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