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  "body": "\n## Statement\n\nLet $C$ be a [[circle|060W5CBZP6]] in the Euclidean plane $\\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $P$ be a point outside $C$.\nLet a [[tangent line|0T7DX8NY1X]] through $P$ touch $C$ at $T$. Let a second line through $P$ meet $C$ at the two points $G_1$ and $G_2$.\nThen\n$$d(P, T)^2 = d(P, G_1)\\, d(P, G_2).$$\n\n## Remarks\n\nBoth sides are equal to the [[power|02E9VT1RVQ]] of $P$ with respect to $C$.\nIt is the limit case of the [[intersecting secants theorem|0J5XR7BW2S]] where one secant becomes a tangent.\nThe theorem is Proposition 36 of Book III of Euclid's Elements.\n\n## Explanations\n\n### L1 human\n\nFrom a point outside a circle, the square of the tangent length equals the product of the two distances along any secant. For the unit circle and $P = (3, 0)$, the secant along the $x$-axis meets the circle at distances $2$ and $4$. So the tangent length is $\\sqrt{8}$.\n",
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