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  "body": "\n## Definition\n\nLet $E$ be a [[Euclidean space|0XPR1E85W0]] of dimension $n$, and fix a [[Cartesian coordinate system|0RWKYX8339]] on $E$.\nIt is a bijection $P \\mapsto x(P) \\in \\R^n$ given by an origin and $n$ mutually perpendicular unit axes.\n*Analytic geometry* is the study of figures in $E$ through their coordinates:\n\n1. a figure $F \\subseteq E$ is described by equations and inequalities, as the set of points $P$ whose coordinates satisfy them;\n2. the [[distance|0NPPGHQNXQ]] between $P$ and $Q$ is computed as $\\abs{x(P) - x(Q)} = \\sqrt{\\sum_{i=1}^n (x_i(P) - x_i(Q))^2}$, and angles through the dot product of coordinate vectors;\n3. geometric statements are proved by algebraic computation on these equations.\n\n## Remarks\n\nExample: in the plane, the line $y = x$ and the circle $x^2 + y^2 = 2$ meet where $2x^2 = 2$, that is at $(1, 1)$ and $(-1, -1)$.\nThe distance between these points is $\\sqrt{2^2 + 2^2} = 2\\sqrt{2}$, the diameter of the circle, as it should be for a line through the centre.\nNon-example: with oblique axes at an angle of $60^\\circ$ and unit lengths on both axes, the formula of clause 2 fails.\nThe points with coordinates $(1, 0)$ and $(0, 1)$ are then at distance $1$, since with the origin they form an equilateral triangle, and not $\\sqrt{2}$.\nAffine notions such as lines, midpoints and parallelism can be studied with any affine coordinates; metric notions need Cartesian ones.\nIn the plane, a line is $\\set{(x, y) : ax + by = c}$ with $(a, b) \\ne (0, 0)$, and the circle with centre $(a, b)$ and radius $r$ is $\\set{(x, y) : (x - a)^2 + (y - b)^2 = r^2}$.\n",
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