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  "body": "\n## Statement\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\\mathbb{R}^n$. Let $d$ be the [[Euclidean distance|0NPPGHQNXQ]], and let $\\angle ABC$ be the [[angle at $B$|0GJAJC6TNZ]].\nThen\n$$d(A, C)^2 = d(A, B)^2 + d(B, C)^2$$\nif and only if $\\angle ABC = \\pi / 2$.\n\n## Remarks\n\nThe direction \"if\" is the classical theorem: in a right triangle, the square of the hypotenuse is the sum of the squares of the legs, $a^2 + b^2 = c^2$. The direction \"only if\" is its converse.\nIn vector form: for vectors $u$ and $v$ with $u \\cdot v = 0$, $\\|u + v\\|^2 = \\|u\\|^2 + \\|v\\|^2$.\nThe theorem is the special case $\\gamma = \\pi / 2$ of the [[law of cosines|060ZVZBV3E]].\nMathlib states the equivalence for all points $A$, $B$ and $C$, with its convention for the angle at a repeated point.\n\n## History\n\nThe theorem carries the name of the Greek philosopher Pythagoras. Euclid's Elements proves it as Proposition 47 of Book I, and the converse as Proposition 48.\n\n## Explanations\n\n### L1 human\n\nBuild a square on each side of a right triangle. The two small squares together have the same area as the big square on the longest side. For the sides $3$, $4$ and $5$: $9 + 16 = 25$.\n",
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