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  "body": "\n## Statement\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane.\nLet $O$ and $R$ be the centre and radius of its [[circumcircle|03VZJE6C8C]], and $I$ and $r$ the centre and radius of its [[incircle|07DFG5JYPT]].\nThen the distance $d = OI$ satisfies\n$$d^2 = R(R - 2r).$$\nIn particular $R \\ge 2r$, with equality if and only if the triangle is equilateral.\n\n## Remarks\n\nLet the line $AI$ meet the circumcircle again at $M$, the midpoint of the arc $BC$ not containing $A$.\nAn angle chase shows $MI = MB$, and the law of sines in the triangle $ABM$ gives $MB = 2R \\sin \\frac{A}{2}$.\nAlso $AI = r / \\sin \\frac{A}{2}$, so $AI \\cdot IM = 2Rr$.\nThe power of the point $I$ with respect to the circumcircle is $R^2 - d^2 = AI \\cdot IM$, and the formula follows.\n\nThe inequality $R \\ge 2r$ is Euler's inequality.\nEquality $d = 0$ means that the [[circumcentre|0F98FZK14A]] and the [[incentre|0M9MX8YW0A]] coincide, which happens only for an equilateral triangle.\n\nExample: for the right triangle with sides $3$, $4$ and $5$, the circumradius is $R = \\frac{5}{2}$ and the inradius is $r = 1$.\nThe formula gives $d^2 = \\frac{5}{2} \\cdot \\frac{1}{2} = \\frac{5}{4}$.\nWith $C = (0, 0)$, $A = (4, 0)$, $B = (0, 3)$ one has $O = (2, \\frac{3}{2})$ and $I = (1, 1)$, and indeed $OI^2 = 1 + \\frac{1}{4} = \\frac{5}{4}$.\n\n## Explanations\n\n### L1 human\n\nEvery triangle has a circle through its corners and a circle inside touching its sides.\nEuler's theorem gives the distance between their centres from the two radii alone.\nAs a consequence, the outer circle is always at least twice as large as the inner one.\nThe two centres meet only when the triangle is equilateral.\n\n### L2 human\n\nThe theorem is a basic tool for inequalities in triangle geometry, through $R \\ge 2r$.\nIt also enters the proof of Feuerbach's theorem via the distance between the incentre and the nine-point centre.\nIts converse is a closure statement.\nFor two circles with $d^2 = R(R - 2r)$, every point of the outer circle is a vertex of a triangle inscribed in one and circumscribed about the other.\nThis is the case of triangles in [[Poncelet's closure theorem|XWYXSRM5YS]].\n",
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