Resumen de Pizza again? On the division of polygons into sections with a common origin

Ilya Sinitsky, Moshe Stupel, Marina Sinitsky

• The paper explores the division of a polygon into equal-area pieces using line segments originating at a common point. The mathematical background of the proposed method is very simple and belongs to secondary school geometry. Simple examples dividing a square into two, four or eight congruent pieces provide a starting point to discovering how to divide a regular polygon into any number of equal-area pieces using line segments originating from the centre. Moreover, it turns out that there are infinite ways to do the division. Discovering the basic invariant involved allows application of the same procedure to divide any tangential polygon, as after suitable adjustment, it can be used also for rectangles and parallelograms. Further generalization offers many additional solutions of the problem, and some of them are presented for the case of an arbitrary triangle and a square. Links to dynamic demonstrations in GeoGebra serve to illustrate the main results.