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July 2nd, 2017, 06:29 PM   #1
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Special triangles

A new method that produces right triangles , with angles close 45 deg'

Select a number close to ( 1 + root of 2 ) for example a = 2.4141

b = 0.5 ( a^2 - 1 ) = 2.41394

c = b +1 = 3.41394

a^2 + b^2 = c^2
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July 2nd, 2017, 06:57 PM   #2
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No it doesn't. And what does "Select a number close to (1 + root of 2)" have to do with anything? Why not just say, select a number a and b and the result is another arbitrary constant.
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July 2nd, 2017, 07:17 PM   #3
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It's so simple

I choose a = 2.41 and that's all.

b = 0.5 ( a^2 - 1) = 2.40405

c = b + 1 = 3.40405

a^2 + b^2 = c^2
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July 2nd, 2017, 07:24 PM   #4
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Quote:
Originally Posted by aetzbar View Post
A new method that produces right triangles , with angles close 45 deg'

Select a number close to ( 1 + root of 2 ) for example a = 2.4141

b = 0.5 ( a^2 - 1 ) = 2.41394 THIS IS NOT CORRECT

c = b +1 = 3.41394 SO THIS IS NOT CORRECT

a^2 + b^2 = c^2 SO THIS ALSO IS NOT CORRECT
I am not sure why we are supposed to pay attention to a post where the arithmetic is not correct. But even if the arithmetic were correct, what would be the significance of finding a right triangle where the other two angles are not equal to 45 degrees? There is nothing untoward with a right triangle having angles of 44, 46, and 90 degrees.

I went to Columbia in the 60's, when you could drink in New York at 18. I heard far more cogent discussions in the freshman dorm after the bars closed.
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July 2nd, 2017, 07:24 PM   #5
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We don't need your method.
Quit being silly...
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July 2nd, 2017, 07:38 PM   #6
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The perfect triangle

Quote:
Originally Posted by JeffM1 View Post
I am not sure why we are supposed to pay attention to a post where the arithmetic is not correct. But even if the arithmetic were correct, what would be the significance of finding a right triangle where the other two angles are not equal to 45 degrees? There is nothing untoward with a right triangle having angles of 44, 46, and 90 degrees.

I went to Columbia in the 60's, when you could drink in New York at 18. I heard far more cogent discussions in the freshman dorm after the bars closed.
a = 1 + root of 2
b = 0.5 ( a^2 - 1 ) = 1 + root of 2
c = b + 1 = 2 + root of 2

a^2 + b^2 = c^2
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July 2nd, 2017, 08:04 PM   #7
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Originally Posted by aetzbar View Post
I choose a = 2.41 and that's all.

b = 0.5 ( a^2 - 1) = 2.40405

c = b + 1 = 3.40405

a^2 + b^2 = c^2
$a = \sqrt{2} + 1 \implies a^2 = 2\sqrt{2} + 3.$

$b = 0.5(a^2 - 1) = 0.5(2\sqrt{2} + 3 - 1) = \sqrt{2} + 1 \implies \\

b^2 = 2\sqrt{2} + 3.$

$c = b + 1 = \sqrt{2} + 2 \implies c^2 = 4\sqrt{2} + 6.$

$a^2 + b^2 = 2\sqrt{2} + 3 + 2\sqrt{2} + 3 = 4\sqrt{2} + 6.$

$a^2 + b^2 = c^2.$

And we deduce from this what exactly?
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July 2nd, 2017, 09:16 PM   #8
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Quote:
Originally Posted by aetzbar View Post
A new method that produces right triangles , with angles close 45 deg
One of the basic lessons in school is that the isosceles triangle with legs $\sqrt2$ and hypotenuse $1$ has angles of exactly $45^\circ$.

We also have methods for generating every single right-angled triangle with integer sides. Do you really think the world cares about a right-angled triangle with angles "close" to $45^\circ$?

If you are going to use irrational numbers, you really ought to get comfortable with them.
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July 2nd, 2017, 09:36 PM   #9
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The number a (that selected,between zero and infinite)

Determines all the triangle data.

b = 0.5 ( a^2 - 1 )
c = 0.5 ( a^2 - 1 ) + 1

circumference = a^2 + a
aria = 0.25 ( a^3 - a)
height = ( a^3 - a) : (a^2 +1)

angle tg against a = 2a : ( a^2 - 1)
half angle tg against a = 1 : a

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July 2nd, 2017, 10:26 PM   #10
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If b = (a² - 1)/2, a² = 2b + 1, and so a² + b² = b² + 2b + 1 = (b + 1)² = c², satisfying Pythagoras. This isn't new.

As a = 1 + √2 satisfies (a² - 1)/2 = a, choosing a to be "close to" 1 + √2 ensures that b is also "close to" 1 + √2, and so the acute angles of the triangle are close to 45°.

In particular, a = 2.4141 implies b = 2.413939405 and c = 3.413939405. To ensure the triangle has an angle of exactly 90°, the values of b and c shouldn't be rounded to fewer decimal places.
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