# Closest approximated rational triangle

1. Mar 3, 2007

### ktoz

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: Mar 3, 2007
2. Mar 4, 2007

### AlephZero

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.

3. Mar 4, 2007

### ktoz

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: Mar 4, 2007