Search

The following problem has created interest in various high schools and even in some colleges. The proof presented here is designed to give the serious student of plane geometry some insight into techniques that can be used in other originals.

The law of cosines has always frustrated those teachers who wish to use it computationally. The transformation developed in this paper makes it more amenable to the use of logarithmic computation.

(9) Several boundary agreements were found by all three approaches. All of them isolated certain heavy industry areas, a few residential neighborhoods, and some large outlying sectors. (10) The social and demographic techniques also singled out small, highly homogeneous, and distinctive areas of the city which were concealed by the ecological approach. For example, Figure 4 shows a segment of the city which is completely enclosed by rivers, railroads, factories, and main thoroughfares. The boundaries indicated by the demographic and social indices do not correspond to the ecological ones. Further, this area is not internally homogeneous from other points of view. Two types of demographic areas have boundaries which at places coincide, fall within, and extend beyond the ecological boundaries. The same applies for two types of social intimacy which lie across the area in question. Obviously compromises among the boundaries were necessary to arrive at a satisfactory general sub-area plan for the city. Since no one of the three plans was satisfactory in every way to meet census tract requirements, they were all used in making the eventual plan. How the compromises were made will be reported elsewhere.