A New Proof of Pappus’s Theorem
نویسنده
چکیده
The realizability problem for rank 3 oriented matroids (see [1]) is equivalent to the pseudoline stretchability problem (see [4]). This paper uses an example to illustrate a new approach to this problem. The main theorem of this paper, like the trigonometric form of Ceva’s theorem, shows a non-trivial relationship amongst the angles in a specific line arrangement figure (c). We work in polar coordinates, with the origin always in the open region of the Euclidean plane, between the last and the first lines (under the polar orientation). To describe those parts of (c) that are significant (the nine shaded triangles, and the polar origin (c′)) we use the notation: (i, j, k) to show that the lines labeled i and k cross on the far side of line j, and (i, j, k)− to show that they cross on the same side of j as the origin. The use of either statement also indicates that 0 < θj − θi, θk − θj , θk − θi < 180.
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