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  "body": "\n## Statement\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane that is not equilateral.\nThen its [[nine-point circle|033H8MQRC3]] is internally tangent to its [[incircle|07DFG5JYPT]].\nIt is also externally tangent to each of its three excircles.\nAn excircle is a circle outside the triangle tangent to one side and to the extensions of the other two sides.\n\n## Remarks\n\nLet $R$ be the circumradius, $r$ the inradius, and $N$ and $I$ the centres of the nine-point circle and of the incircle.\nThe nine-point circle has radius $R/2$, and $N$ is the midpoint of the circumcentre $O$ and the [[orthocentre|0WB619CA0V]] $H$.\nWrite $a, b, c$ for the side lengths, and $A, B, C$ also for the angles.\nTwo known formulas give $OH^2 = 9R^2 - (a^2 + b^2 + c^2)$ and $IH^2 = 2r^2 - 4R^2 \\cos A \\cos B \\cos C$.\nWith these and [[Euler's theorem|RFCQT6TKZF]] for $OI$, [[Apollonius's theorem|0NAKHQ9XYY]] in the triangle $OIH$ gives $IN = R/2 - r$.\nThis says the two circles are internally tangent; a similar computation gives $I_a N = R/2 + r_a$ for each excentre $I_a$ with exradius $r_a$.\n\nIn an equilateral triangle the nine-point circle and the incircle coincide, since $N = I$ and $R/2 = r$.\nThe point of tangency with the incircle is the Feuerbach point of the triangle.\n\n## Explanations\n\n### L1 human\n\nEvery triangle has a special circle through nine notable points, the nine-point circle.\nIt also has an inscribed circle and three circles that touch its sides from outside.\nFeuerbach's theorem says the nine-point circle touches all four of these circles.\nThis is a surprising coincidence that holds for every triangle.\n\n### L2 human\n\nThe theorem is a highlight of classical triangle geometry, published by Karl Wilhelm Feuerbach in 1822.\nIt ties together the [[circumcircle|03VZJE6C8C]], the incircle and the nine-point circle through the radii $R$, $r$ and $R/2$.\n",
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  "name": "Feuerbach's theorem",
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