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  "body": "\n## Statement\n\nWork in the real [[projective plane|DANFNN8C4Y]] $\\mathbb{P}^2(\\R)$, and let $\\mathcal{C}$ be a nondegenerate [[conic|00HJCMC2CD]].\nLet $t_1, \\dots, t_6$ be six distinct lines [[tangent|01R1HW3Y42]] to $\\mathcal{C}$.\nPut $A = t_1 \\cap t_2$, $B = t_2 \\cap t_3$, $C = t_3 \\cap t_4$, $D = t_4 \\cap t_5$, $E = t_5 \\cap t_6$ and $F = t_6 \\cap t_1$.\nThen the three lines $AD$, $BE$ and $CF$ pass through a common point.\n\n## Remarks\n\nSo $ABCDEF$ is a hexagon whose sides touch the conic, and the statement says that its three main diagonals are concurrent.\nNo three tangent lines of a nondegenerate real conic pass through one point, which makes the six vertices distinct and the three diagonals well defined.\n\nThe theorem is the dual of [[Pascal's theorem|94AKWRHND0]].\nThe tangent lines of a nondegenerate conic form a nondegenerate conic of the dual plane, and the dual of the statement \"three points are collinear\" is \"three lines are concurrent\".\nSo the principle of [[duality|DCMP1A0138]] applied to Pascal's theorem gives Brianchon's theorem.\n\nExample: for a regular hexagon circumscribed about a circle, the three main diagonals pass through the centre of the circle.\n\n## Explanations\n\n### L1 human\n\nDraw six lines that just touch an ellipse, so that they enclose a hexagon around it.\nJoin opposite corners of the hexagon by three diagonals.\nThese three diagonals always meet at a single point.\n\n### L2 human\n\nBeing the dual of Pascal's theorem, it is a classic illustration of projective [[duality|DCMP1A0138]].\nIt is used to construct tangent lines of a conic from five given tangent lines with a straightedge.\nThe same argument works over any field of characteristic different from $2$.\nIn characteristic $2$ all tangent lines of a nondegenerate conic pass through one point, so the hypotheses of the statement cannot be met there.\n",
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