Two years ago I came across a very beautiful geometric construction (Figure 1), on a tee-shirt I received from the Upstate New York math team. Written on the shirt was a surprising result of iterating a certain construction multiple times. The coaches told me that the proof for the result which they had was very long and very complex. Such a beautiful problem deserves a beautiful answer. So, I came up with a very nice proof which took less than a minute to present to the team the following year. Since then I have delved further into the construction, arriving at some very interesting results. This paper proves some of those results which arise from iterating the geometric construction.