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.