Circles and Squares
First I constructed a square and circumscribed it with a circle.
Next I constructed the incircle, using the midpoints of the sides of the square.
In order to come up with a conjecture I calculated the areas of each
of the Circles and the square
I found the perimeter of the square and the Circumference of the circles.
I compared the ratios of the Circumferences to Perimeter, and the
ratio of the Areas of the circles to the area of the square. I found
some very uninteresting numbers. The last thing that I checked out
was the ratio of the Areas of the two Circles. I found that the area
of the Incircle is half of the area of the Cirmscribed circle.
You can drag one of the points and show that no matter the size of
the square and it's appropriate circles the ratio holds true. Why?...
...If we break down just the square we find some interesting relationships
with the triangles.
So, if you construct another square inside of the given incircle, the
area of that incircle would be half the given. Hmmmmm?......