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  "body": "\n## Statement\n\nLet $A$, $B$ and $C$ be points of the Euclidean space $\\mathbb{R}^n$, and let $D$ be the [[midpoint|0S706JBZBG]] of $B$ and $C$. Let $|XY|$ denote the [[distance|0NPPGHQNXQ]] between the points $X$ and $Y$.\nThen\n$$|AB|^2 + |AC|^2 = 2 \\left( |AD|^2 + |BD|^2 \\right).$$\n\n## Remarks\n\nFor a triangle $ABC$ with the side lengths $a = |BC|$, $b = |CA|$, $c = |AB|$, the [[median|0XW8HX8VEV]] from $A$ thus has the length $m_a = \\frac{1}{2} \\sqrt{2 b^2 + 2 c^2 - a^2}$.\nIn vector form, with $u = B - A$ and $v = C - A$, the statement is the parallelogram law $\\|u + v\\|^2 + \\|u - v\\|^2 = 2\\|u\\|^2 + 2\\|v\\|^2$.\nFor an isosceles triangle with $|AB| = |AC|$, the median $AD$ is perpendicular to $BC$, and the theorem reduces to the [[Pythagorean theorem|0VP3A1290K]].\n\n## History\n\nThe theorem is Proposition 122 of Book VII of the Collection of Pappus of Alexandria. It was possibly in the lost work Plane Loci of Apollonius of Perga.\n\n## Explanations\n\n### L1 human\n\nThe theorem relates the two sides from $A$ to the median from $A$. If you know the three sides of a triangle, it gives the length of each median. For the right triangle with legs $3$, $4$ and hypotenuse $5$, the median to the hypotenuse has length $\\frac{1}{2}\\sqrt{18 + 32 - 25} = \\frac{5}{2}$.\n",
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