Triangles of Equal Area and Perimeter and Inscribed Circles

Abstract

Much has been written on triangles having areas and perimeters of equal numerical value. Markowitz (1981) determined that only five triangles having sides with integral lengths exist for which the areas equal the perimeters, but that infinitely many have rational side lengths. (Throughout this article, when it is stated that triangles have equal area and perimeter, it means that they are numerically equal.) Markowitz (1981) and Bates (1979) both determine equations satisfied by triangles having equal areas and perimeters, but these equations can only be used through trial and error. The purpose of this article is to find a method that will easily generate any number of triangles of equal area and perimeter, and also geometrically characterize such triangles.