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  "body": "\n## Statement\n\nLet $k$ be a [[field|04C5ZBG1WT]], and work in the [[projective plane|DANFNN8C4Y]] $\\mathbb{P}^2(k)$.\nLet $\\mathcal{C}$ be a nondegenerate [[conic|00HJCMC2CD]]: the zero set of a homogeneous quadratic polynomial $F$ in three variables.\nHere $F$ is not a product of linear factors over any extension of $k$.\nLet $A, B, C, D, E, F$ be six distinct points of $\\mathcal{C}$.\nThen the three points\n$$AB \\cap DE, \\quad BC \\cap EF, \\quad CD \\cap FA$$\nare [[collinear|0AFNNTRTJE]].\n\n## Remarks\n\nHere $AB$ denotes the line through $A$ and $B$.\nNo three points of a nondegenerate conic are collinear, so the lines in each intersection are distinct and the three points are well defined.\nThe line through them is the Pascal line of the hexagon $ABCDEF$.\n\nOne proof over $\\C$, which extends to any field by passing to an algebraic closure, uses cubic curves.\nThe cubics $AB \\cdot CD \\cdot EF$ and $BC \\cdot DE \\cdot FA$, products of three lines each, meet in the six points and in the three points of the statement.\nA suitable combination of them vanishes on a seventh point of $\\mathcal{C}$, so by Bézout's theorem it contains $\\mathcal{C}$, and the remaining factor is a line through the three points.\n\nExample: on the unit circle $x^2 + y^2 = 1$ in the real plane, take a regular hexagon.\nOpposite sides are then parallel, so the three points lie on the line at infinity.\n\nWhen the conic degenerates into two lines, the statement becomes [[Pappus's hexagon theorem|B6C3J2GJWP]].\nThe dual statement is [[Brianchon's theorem|3THE3Q96KK]].\n\n## Explanations\n\n### L1 human\n\nPick six points on an ellipse, number them around the curve or in any order, and join them into a closed hexagon.\nExtend the three pairs of opposite sides until each pair crosses.\nThe three crossing points always lie on one straight line.\n\n### L2 human\n\nPascal's theorem is a basic tool for constructions with conics: given five points of a conic, it lets one construct any number of further points with a straightedge alone.\nIts converse also holds: if the three points are collinear, the six points lie on a conic, which may be degenerate.\nIt is the projective form of a statement that, in the real plane, covers ellipses, parabolas and hyperbolas at once, since these are all the same in projective geometry.\nThrough [[duality|DCMP1A0138]] it gives [[Brianchon's theorem|3THE3Q96KK]].\n",
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