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  "body": "\n## Definition\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].\nThe *Fermat point* of $ABC$ is the unique point $F$ of the plane that minimises\n$$f(P) = d(P, A) + d(P, B) + d(P, C).$$\n\n## Remarks\n\nA minimiser exists because $f$ is continuous and tends to infinity far away; it is unique because $f$ is strictly convex when $A$, $B$, $C$ are not collinear.\nIf each angle of $ABC$ is less than $120^\\circ$, then $F$ is the interior point with $\\angle AFB = \\angle BFC = \\angle CFA = 120^\\circ$.\nIn that case $F$ is also the common point of the lines that join each vertex to the apex of the outward equilateral triangle on the opposite side.\nIf the angle at a vertex is at least $120^\\circ$, then $F$ is that vertex.\nThe point is also called the Torricelli point.\n\nExample: in an equilateral triangle the centre $O$ sees each side under $120^\\circ$ by symmetry, so $F = O$.\n\nNon-example: take $A = (0, 0)$, $B = (1, 0)$, $C = (0, 1)$ and the [[centroid|0X7EEWES5G]] $G = (1/3, 1/3)$.\nThen $f(G) = (\\sqrt 2 + 2\\sqrt 5)/3 \\approx 1.962$.\nAll angles are below $120^\\circ$, and the minimum of $f$ is $\\sqrt{2 + \\sqrt 3} \\approx 1.932$, so $G$ is not the Fermat point.\n",
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