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  "body": "\n## Definition\n\nLet $T$ be the set of triples $(a, b, c)$ of positive real numbers with $a \u003c b + c$, $b \u003c c + a$ and $c \u003c a + b$.\nThese are the side lengths of the [[triangles|0CTNFGYMM2]] of the Euclidean plane.\nA *triangle center function* is a real function $f$ on $T$, not identically zero, such that:\n\n1. $f$ is homogeneous: there is a real $n$ with $f(ta, tb, tc) = t^n f(a, b, c)$ for all $t \u003e 0$ and $(a, b, c) \\in T$.\n2. $f$ is bisymmetric: $f(a, c, b) = f(a, b, c)$ for all $(a, b, c) \\in T$.\n\nLet $ABC$ be a triangle with $a = BC$, $b = CA$ and $c = AB$.\nThe *triangle center* given by $f$ is the point of the plane with trilinear coordinates\n$$f(a, b, c) : f(b, c, a) : f(c, a, b),$$\nwhenever these are not all zero.\n\n## Remarks\n\nTrilinear coordinates $\\alpha : \\beta : \\gamma$ describe the point whose signed distances to the lines $BC$, $CA$, $AB$ are in the ratio $\\alpha : \\beta : \\gamma$.\nHomogeneity makes the point move with the triangle under scaling, and bisymmetry makes it independent of the labelling of the vertices.\n\nExample: $f(a, b, c) = 1$ is homogeneous of degree $0$ and bisymmetric.\nIts center $1 : 1 : 1$ is equidistant from the three sides, so it is the [[incentre|0M9MX8YW0A]].\nSimilarly $f(a, b, c) = 1/a$ gives the [[centroid|0X7EEWES5G]].\n\nNon-example: $f(a, b, c) = c/b$ is homogeneous but not bisymmetric, since $f(a, c, b) = b/c$.\nThe point $c/b : a/c : b/a$ is the first Brocard point, which changes into the second Brocard point when two vertices are swapped, so it is not a triangle center.\n",
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