{
  "aliases": [
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  "body": "\n## Statement\n\nLet $A$, $B$, $C$ and $D$ be [[concyclic points|049DCJZ448]] of the Euclidean plane $\\mathbb{R}^2$, with $A \\neq D$ and $B \\neq C$. Let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].\nSuppose that the lines $AD$ and $BC$ meet at a point $P$ outside the circle.\nThen\n$$d(P, A)\\, d(P, D) = d(P, B)\\, d(P, C).$$\n\n## Remarks\n\nBoth products are equal to the [[power|02E9VT1RVQ]] $d(P, O)^2 - r^2$ of $P$, where $O$ is the centre and $r$ the radius.\nA proof: the triangles $PAC$ and $PBD$ are similar, because they have the same angle at $P$ and equal inscribed angles at $C$ and $D$.\nIf the second line becomes tangent, then $B = C$, and the statement becomes the [[tangent–secant theorem|0QX7E9KAAX]].\n\n## Explanations\n\n### L1 human\n\nFrom a point outside a circle, draw two lines that cut the circle. On each line, multiply the distances from the point to the two crossing points. The two products are equal.\n",
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