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Abstract. The so-called "Japanese theorem" dates back over 200 years; in its original form it states that given a quadrilateral inscribed in a circle, the sum of the inradii of the two triangles formed by the addition of a diagonal does not depend on the choice of diagonal. Later it was shown that this invariance holds for any cyclic polygon that is triangulated by diagonals. In this article we examine this theorem closely, discuss some of its consequences, and generalize it further.

In the United States, congressional districts are redrawn every ten years based on changes in population revealed by the census. Individual states are responsible for redrawing their congressional districts. Often sophisticated (and expensive) software packages are used to guide redistricting committees when drawing the new boundaries. Much of the cost is due to the fact that redistricting is a fantastically complicated problem. We do not propose to give a definitive way of building political districts.

This article explores analogues of the Pythagorean Theorem in non-Euclidean geometries.

This article analyzes the physical and mathematical properties of the mirascope and models the mirascope using dynamic learning technology