Cyclotomic concurrency
This article deals with the study of the possible intersection in one interior point of three diagonals of a regular polygon. It starts by giving an arithmetic characterization of the problem and presenting the proof of some elementary theorems that give some criteria for the intersection in one point of three cords of a circle. The author is interested in counting how many concurrences exist in \(n\)-sided polygons. In order to characterize really distinct concurrences he defines equivalent concurrences as those which can be transformed into each other by concurrence transformations (isometries of the circle). This definition permits the characterization of a representative of each class in a nice way. Since a regular polygon can be inscribed in a circle it is then possible to rewrite the criterion presented for circles for the special representative of each class. NEWLINENEWLINENEWLINEIn the fifth section the author considers the case of a regular polygon with an even number of sides and considers all dichotomic concurrences defined by the diagonals proving two results that give the total number of classes of existing equivalent concurrences. Section six is concerned with the case of non-dichotomic concurrences for a general regular polygon. It starts with similar results as the previous section with more elaborate calculations. In order to simplify these computations the author introduces cyclotomic polynomials. Since the formula first introduced in this section is a null linear combination of roots of unity it can be further simplified. In the last part of section six one can find a table with non-dichotomic concurrences for regular \(n\)-sided polygons, for \(n<200.\)NEWLINENEWLINENEWLINEIn section seven it is given a necessary and sufficient condition for the concurrence of three cords with endpoints on vertices of a polygon to be cyclotomic. The article finishes with an interpretation of this problem in projective geometry and in the very last section with the concurrence in an exterior point.NEWLINENEWLINENEWLINEThroughout the article the author explains a great deal of examples were most of the theorems proved are applied.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00066].
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