Pappus定理的对偶

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这个资料第7页说Pappus定理的对偶是

The dual of Pappos' theorem states that beginning with an arbitrary pair of points labeled $A$ and $B$, consider a pencil of three lines labeled $r_1, b_1, y_1$ incident to $A$ and $r_2, b_2, y_2$ incident to $B$. The intersections of $r_1$ with $b_2$ and $b_1$ with $r_2$ defines a line labeled $p$. The intersections of $r_1$ with $y_2$ and $y_1$ with $r_2$ defines a second line labeled $o$. The intersections of $b_1$ with $y_2$ and $y_1$ with $b_2$ defines a third line labeled $g$. The three lines $p,o,g$ meet at a third point $C$ which mysteriously appears.

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Wikipedia也有一个dual theorem of Pappus和1楼的相同

If 6 lines $ A,b,C,a,B,c $ are chosen alternately from two pencils with centers $ G,H $, the lines\begin{gathered}X:=(A\cap b)(a\cap B)\\Y:=(c\cap A)(C\cap a)\\Z:=(b\cap C)(B\cap c)\end{gathered}are concurrent, that means: they have a point $ U $ in common.

这和原定理不同吧

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MathWorld

Pappus's hexagon theorem is self-dual. The Levi graph of the $9_3$ configuration corresponding to the theorem is the Pappus graph.

cut-the-knot

The second remarkable feature of the configuration is that it is self-dual.

Wikipedia

This configuration is self dual. Since, in particular, the lines $ Bc,bC,XY $ have the properties of the lines $ x,y,z $ of the dual theorem, and collinearity of $ X,Y,Z $ is equivalent to concurrence of $ Bc,bC,XY $, the dual theorem is therefore just the same as the theorem itself. The Levi graph of the Pappus configuration is the Pappus graph, a bipartite distance-regular graph with 18 vertices and 27 edges.