engleski

On equations and properties concerning some classes of chordal polygons

In the paper $k$-chordal (or $k$-inscribed) polygons of first and second kind with given index are considered. Existence result is proved for equilateral chordal polygon which side lengths are already known. The convex and nonconvex cases are discussed depending on the orientation of the polygon. Secondly, the number of different radii of circumcircles of equilateral $k$-inscribed $n$-gon cannot be greater then \vspace{; ; ; ; 2mm}; ; ; ; \[ \mathbf s[n] = \left[ \frac{; ; ; ; n-1}; ; ; ; {; ; ; ; 2}; ; ; ; \right] + \left[ \frac{; ; ; ; n-3}; ; ; ; {; ; ; ; 2}; ; ; ; \right] +\left[ \frac{; ; ; ; n-5}; ; ; ; {; ; ; ; 2}; ; ; ; \right] + \cdots + 2+ 1 . \] Very natural conjecture is formulated on the existence of side lengths of $k$-chordal $n$-gons when the minimal number of different circumcircle radii is \vspace{; ; ; ; 2mm}; ; ; ; \begin{; ; ; ; align*}; ; ; ; {; ; ; ; \bf \boldsymbol{; ; ; ; \sigma}; ; ; ; }; ; ; ; [n] &:= \left[ \frac{; ; ; ; n-1}; ; ; ; {; ; ; ; 2}; ; ; ; \right] + \binom n1 \left[ \frac{; ; ; ; n-3}; ; ; ; {; ; ; ; 2}; ; ; ; \right] + \binom n2 \left[ \frac{; ; ; ; n-5}; ; ; ; {; ; ; ; 2}; ; ; ; \right] + \cdots \\ &\cdots + \binom n\mu \left[ \frac{; ; ; ; n-2\mu+1}; ; ; ; {; ; ; ; 2}; ; ; ; \right], \end{; ; ; ; align*}; ; ; ; where $n-2\mu = 3 (4)$ for $n$ odd (even). Thirdly, the so-called {; ; ; ; \em main equation}; ; ; ; (kind of related characteristic algebraic equation for a polygon) is introduced for the class $C_n(a_1, \cdots, a_n)$ of $k$-chordal {; ; ; ; \em related polygons}; ; ; ; . In few illustrative examples we obtain the number and the numerical values of different radii of quadrangle, pentagon, octagon and enneagon, solving the related main equations, when only the side lengths of initial polygons are known. In the final section certain interesting properties of the so-called main equations are discussed, proving that the positive roots of the main equations are the radii of the circumcircles of the chordal $n$-gons whose sides have the lengths $a_1, \cdots, a_n$. The equilateral pentagon is presented in detail with three different positive solutions of its main equation which is an eigth degree algebraic equation. In the same section the main equation of $\lambda n$-gons is characterized, when the initial $n$-gon is $\lambda$ times continued on the same circumcircle, $\lambda$ positive integer.

Algebraic equations; $k$-chordal polygon; $k$-inscribed chordal polygon; main equation; circumcircle; polyogn of first kind; polygon of second kind; index of chordal polygon

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