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  "body": "\n## Definition\n\nLet $C$ be a [[circle|060W5CBZP6]] in the Euclidean plane $\\mathbb{R}^2$ with the centre $O$ and the radius $r$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].\nThe power of a point $P \\in \\mathbb{R}^2$ with respect to $C$ is the real number\n$$\\Pi(P) = d(P, O)^2 - r^2.$$\n\n## Remarks\n\n$\\Pi(P) \u003e 0$ if $P$ is outside $C$, $\\Pi(P) = 0$ if $P$ is on $C$, and $\\Pi(P) \u003c 0$ if $P$ is inside $C$.\nFor $P$ outside $C$ and a tangent from $P$ that touches $C$ at $T$, the Pythagorean theorem gives $\\Pi(P) = d(P, T)^2$.\nLet a line through $P$ meet $C$ at $S_1$ and $S_2$. Then $d(P, S_1)\\, d(P, S_2) = |\\Pi(P)|$. This contains the [[intersecting chords theorem|0YQQYET8YQ]] and the [[intersecting secants theorem|0J5XR7BW2S]].\nJakob Steiner introduced the power of a point in 1826.\n\n## Explanations\n\n### L1 human\n\nThe power of a point measures how far the point is from a circle, in a squared form. For the unit circle and $P = (3, 0)$, the power is $9 - 1 = 8$. A tangent from $P$ to the circle has the length $\\sqrt{8}$.\n",
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