ABSTRACT
A difŠcult problem in geometry can often be transformed into a much simpler one by use of a geometry-preserving mapping in the plane. When this is possible, conclusions are made for the new simpliŠed problem that are invariant under the mapping, and these conclusions are then transformed back to the original problem using the inverse mapping, thus solving the problem. For a nongeometric example, suppose we wanted to Šnd the product of 358 and 762 in base 9. Since multiplication in base 9 is rather nebulous, we Šrst change the numbers to base 10, Šnd the product (which can be done on a pocket calculator), then transform the answer back to base 9. (In case you want to try this, the answer is 310857, base 9.)
The general class of transformations most useful in geometry would presumably be those which preserve collinearity (thus mapping lines to lines). Special subclasses would include mappings which preserve angle measure or distance. A systematic study of these transformations will be undertaken here. Although this theory is interesting in itself, its applications in solving geometry problems is our main goal.
