Closest approximated rational triangle

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The discussion centers on the challenge of finding the closest approximated rational triangle given two points on a plane and a third point that can be placed arbitrarily. Participants conclude that while one can define a search area around the third point, the nature of rational numbers means that there is always a closer approximation possible. A redefinition of the problem is proposed, focusing on rational sides defined by natural numbers and a unit fraction, which may provide a more structured approach to the approximation challenge.

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Given 2 points on a plane, if you arbitrarily place a third, is there any way to determine the closest approximation to this triangle where all sides of the approximation are rationally related?

The only thing I can think of would be to draw a small circle around the third point that represents the acceptable search area for the third point of the approximation, but after that, I'm completely stuck.

Anyone know if there is a fairly efficient way to do this?

Thanks
 
Last edited:
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No, because however close your approximation, there is another one which is closer. Between any two numbers (either real or rational) there are an infinite number of other rational numbers.
 
AlephZero said:
No, because however close your approximation, there is another one which is closer. Between any two numbers (either real or rational) there are an infinite number of other rational numbers.

Hmmm. What if the problem is redefined to:

The closest approximation whose sides are related by:

x, y, z, n = element of natural numbers
x, y, z < max
unit = 1/n

a = x * unit
b = y * unit
c = z * unit
 
Last edited:

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