Statement from my earlier post "The relative length of the height is simply N x 2 'units'."
Bayard's comment: "I can't see from where you derive that."
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Easy enough, it's a bog standard quadratic/algebra slog.
You are told, for example, that the height of a right triangle is one seventh of the total perimeter.
Start off by writing "n = 7" and "N = 6" as before.
As we are looking at relative values, not absolute values, you can ascribe any value you like to the height to get the ball rolling.
It's easiest starting with height = 1. Therefore, the total length of base + hypotenuse = 6. The base is shorter than the hypotenuse, so as a first approximation, the ratios are 1 to 3-minus-a-bit to 3-plus-the-same-bit. We express these as 1; 3 - x; and 3 + x.
Applying Pythagoras, 1^2 + (3-x)^2 = (3+x)^2.
Expand those to get 1 + 9 - 6x + x^2 = 9 + 6x + x^2.
Subtract the right hand side from the left hand side, shift the -12x over, change the sign and you end up with 1 = 12x.
So you have height = 1; base = 2 11/12 and hypotenuse = 3 1/12.
Then express everything in terms of 1/12's.
Height = 12/12; base = 35/12 and hypotenuse = 37/12.
We are looking at relative not absolute values, so we can define our "unit" as 1/12, or indeed "X", which we now know is 1/12, and we express the relative lengths as:
Height = 12 units; base = 35 units; and hypotenuse = 37 units.
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Our "unit" must come out as 1/2N...
The "3" in (3 - x ) and (3 + x) is simply N/2 (it must be, that's how you set up the equation). When you square (3 - x) and subtract the square of (3 + x) you end with 12x (the 9's and x^2's cancel out), "12" here is just 2N (two lots of twice half of N).
The 12x in turn is therefore = 1/(4 * N/2) or 1/(2N). The "1" here is just the value you ascribed to the height. Whatever n or N is, when you do the workings, you always end up with 4N*x = 1.
When you get rid of the 1/12's and strip it down to whole number relative values (12-35-37), the height ends up as 1/(1/2N) which must be 2N.
That explains why height = 2N.
The base is 1 unit shorter and the hypotenuse is 1 unit longer than N/2 (again, by definition, that's how you set it up), once you multiply up by 2N (to get rid of the fractions), the apparent difference is two units.
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Sorry to ruin the magic for you :-(