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Concurrent Diagonals in a 30-Gon

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When three lines meet at a point, they are concurrent. For the diagonals of a 30-gon, out of the 16801 intersection points, there are 3001 points where three or more lines intersect, with 193 different ways to select three concurrent diagonals.

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Consider six points that divide a circle into arcs . If , then the three lines connecting opposing points are concurrent [1]. This property was used to find the sets of lines. By the pizza theorem, half the circle is covered by wedges , and half by wedges .

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Contributed by: Ed Pegg Jr (November 2015)
Open content licensed under CC BY-NC-SA

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In the initialization, {1,3,9,4,8,5,14} is a sample dataset (selected triple 102). The first six numbers are a partition of 30; the last number is the sector of the 30-gon where the intersection occurs.

References

[1] B. Poonen and M. Rubinstein, "The Number of Intersection Points Made by the Diagonals of a Regular Polygon," arxiv.org/abs/math/9508209.

[2] Wikipedia. "Pizza Theorem." (Nov 13, 2015) en.wikipedia.org/wiki/Pizza_theorem.

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