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  "body": "\n## Statement\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane $\\mathbb{R}^2$. For [[collinear points|0AFNNTRTJE]] $X$, $Y$, $Z$ with $Z \\neq Y$, let the signed ratio $\\frac{XZ}{ZY}$ be the real number $\\lambda$ with $Z - X = \\lambda\\,(Y - Z)$.\nLet $\\ell$ be a [[line|0MP9BCF6DP]] that does not pass through $A$, $B$ or $C$. Suppose that $\\ell$ meets the lines $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$.\nThen\n$$\\frac{AF}{FB} \\cdot \\frac{BD}{DC} \\cdot \\frac{CE}{EA} = -1.$$\nConversely, let $D$, $E$ and $F$ be points of the lines $BC$, $CA$ and $AB$, different from the vertices, with this product equal to $-1$. Then $D$, $E$ and $F$ are collinear.\n\n## Remarks\n\nWith unsigned lengths, the product is $1$. The converse of this weaker form is false in general.\nThe formula differs from the formula of [[Ceva's theorem|0KN93VDSXA]] only in the sign.\nThe line $\\ell$ meets the inside of exactly two sides, or of no side. Thus one or three of the ratios are negative, and the product is negative.\n\n## History\n\nThe oldest surviving statement is in the Spherics of Menelaus of Alexandria. There, he uses the plane version as a lemma for a version on the sphere.\nPtolemy used the theorem in the Almagest for problems of spherical astronomy.\n\n## Explanations\n\n### L1 human\n\nA straight line that cuts across a triangle meets the three side lines in three points. These points cut the sides in certain ratios. Going round the triangle, the product of the three signed ratios is always $-1$.\n",
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  "msc": [
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  "name": "Menelaus's theorem",
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