Hi Mat. Not to be condescending but like Mick said, this is basic high school math and geometry. I tutored math in college so I'll try to show you why it's trivial and why the areas of the circles are just a function of basic ratios described by really really old formulas for the polygons and not some kind of new discovery.

I'll review "theorems" 2 and 3 because they're super easy to understand and that's about all I have time for right now.

Theorem 2 says the areas of an equilateral triangle's circumscribed and inscribed circles have a 4:1 ratio. Here are the relevant formulas for an equilateral triangle:

The radius of the circumscribed circle or the length from the triangles center to a corner is R = square root of 3 divided by 3, times the length of a side of the triangle.

The radius of the inscribed circle or the length from the triangles center on a line perpendicular to a side is r = square root of 3 divided by 6, times the length of a side of the triangle.

Notice the difference in the formulas is for R you divide by 3 but for r you divide by 6. That comes to 1/3 versus 1/6. 1/6 is half of 1/3 meaning the distance from the incenter to a corner (R) is twice as long as the distance from the incenter to a side (r) for a ratio of 2:1. The formula for the area of a circle is pi times radius squared. So if one radius R = 2 and the other radius r = 1, then 2 squared = 4 and 1 squared = 1 for a ratio of 4:1. Because of the formula for area of a circle is dependent on the radius squared, it's simply a matter of squaring the ratio of 2:1 for the radii to equate to the ratio 4:1 for the areas.

Theorem 3 states the areas of a square's circumscribed and inscribed circles have a 2:1 ratio. Here are the relevant formulas for a square.

The radius of the circumscribed circle or from the center of the square to a corner is R = half the length of a side times the square root of 2.

The radius of the inscribed circle or from the center of the square on a line perpendicular to a side is r = half the length of a side.

Since half the length of a side is the same in both formulas they cancel out for a value of 1 so the ratio of R:r is the square root of 2:1.

Again, the formula for area of a circle is pi times radius squared. The square root of 2 squared = 2 and the square root of 1 = 1 for a ratio of the areas of the circles at 2:1

The link I provided previously demonstrates Mick and I aren't the only ones that see through Hawkin's ruse:

http://www.skepticforum.com/viewtopic.php?f=7&t=13626
The only reason I can figure why nobody responded to Hawkins is that his claims of new or undiscovered Euclidean geometry are so blatantly false it's self evident to anyone who understands high school geometry such that any response would just be a waste of time. Speaking of which....