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  "body": "\n## Statement\n\nLet $C$ be a [[circle|060W5CBZP6]] with the centre $O$ and the radius $r \u003e 0$ in the Euclidean plane $\\mathbb{R}^2$. Let $T$ be a point of $C$, and let $\\ell$ be a line through $T$.\nThen $\\ell$ is [[tangent|0T7DX8NY1X]] to $C$ at $T$ if and only if $\\ell$ is perpendicular to the radius $OT$. That is, $(X - T) \\cdot (T - O) = 0$ for each point $X$ of $\\ell$.\n\n## Remarks\n\nA proof: for $X = T + t v$ on $\\ell$, we have $\\|X - O\\|^2 = r^2 + 2t\\, v \\cdot (T - O) + t^2 \\|v\\|^2$. So $t = 0$ is the only point of $\\ell$ on $C$ if and only if $v \\cdot (T - O) = 0$.\nThus each point of $C$ has exactly one tangent line.\nThe same holds for a sphere and its tangent plane in $\\mathbb{R}^3$.\n\n## Explanations\n\n### L1 human\n\nTo draw the tangent to a circle at a point, first draw the radius to that point. The tangent is the line through the point at a right angle to the radius. For the unit circle at $(1, 0)$, the radius is horizontal and the tangent is the vertical line $x = 1$.\n",
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        "locator": "Tangent lines to one circle",
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