On algebraic equations concerning semi-tangential polygons (Q2746397)

If \(A_1,\dots,A_n\) are any given different points in a plane, then the union \(A_1A_2 \cup A_2 A_3 \cup \cdots \cup A_{n-1} A_n \cup S\) of line segments \(A_1A_2,\dots,A_{n-1} A_n\) and the set \(S\) which is either the empty set or the segment \(A_nA_1\) is called a semi-polygon and denoted by \(A_1 \cdots A_n\). A semi-polygon \(A_1 \cdots A_n\) will be called a tangential semi-polygon if there is a cycle \(C\) such that each side of the semi-polygon lies on a tangent line of \(C\) and, in case \(A_1 \cdots A_n\) is not a polygon, the end-vertices \(A_1\) and \(A_n\) lie on \(C\). NEWLINENEWLINENEWLINEIn the paper, the equation is derived for the radius of a semi-polygon with given lengths for the tangents. Some properties of this equation are determined, e.g., the number of positive solutions. NEWLINENEWLINENEWLINEIt should be mentioned that the paper contains several inaccuracies and that some results are expected if one keeps in mind the geometrical interpretation.