An isoperimetric theorem in plane geometry

Let P be a simple polygon. Let the vertices of P be mapped, according to a counterclockwise traversal of the boundary, into a strictly increasing sequence of real numbers in [0, 2π). Let a ray be drawn from each vertex so that the angle formed by the ray and a horizontal line pointing to the right equals, in measure, the number mapped to the vertex. Whenever the rays from two consecutive vertices intersect, let them induce the triangular region with extreme points comprising the vertices and the intersection point. It is shown that there is a fixed α such that if all of the assigned angles are increased by α the triangular regions induced by the redirected rays cover the interior of P. This covering implies the standard isoperimetric inequalities in two dimensions, as well as several new inequalities, and resolves a question posed by Yaglom and Boltanskiǐ.