Circle Inversions and Applications

There were a lot of geometric discoveries made in the 19th century that not many people know much about. By "not many" I mean not only those who end their mathematical studies in high school or even college, but even many mathematicians who spend their limited time and energy in other areas. Thanks to some though we do have some excellent texts that capture some of these results...one being Dan Pedoe's "Geometry A Comprehensive Course". In the last entry (Dec 7, 2017) we gave a detailed answer to an introductory exercise from Pedoe's book which opened up a whole universe of interrelated circles. Since then, having delved further into this book, this blogger learned about a 19th century geometric tool that gives insight not only regarding that problem we analyzed using primitive means, but offers a way to solve many problems in a way that we might just not be able to get our heads around using only primitive tools. This "new" tool is that of "inversion in a circle". These inversions transform circles and lines in the planes into other circles and lines while keeping the angles of intersection of any two such figures (and thus tangency and orthogonality).

In the paper Circles: Inversions and Applications , we have presented an introduction to this topic, proved the necessary results needed to establish the tools and worked many examples and exercises using the tools. Taken in its entirety it will give the reader a firm grasp of inversion as a tool to use in everyday geometric situations.

This blogger wants to send kudos to Dan Pedoe for his excellent book of which this is but a small slice.

It is a shame that the standard mathematical curriculum jumps away from geometry so soon when there is so much more to explore. I have run into many people who say they loved Geometry but began to lose interest in subsequent courses (how might that be possible?). Although the Greeks did amazing work in Geometry it should be no surprise that the subsequent centuries would add to that heritage. Well, here at oriolescience we like to offer up a bit of what you may have missed.

One solution we explain in detail in the paper is the Steiner Problem. We give the theoretical solution and work out a detailed example. This in itself is worth the reader's time and attention.