How can we get the approximate values $a,b,c$?

Question

How can we get the approximate values $a,b,c$?

The condition and relation are the followings :

1. $0 < a,b,c < 1$

2. $a + b + c = 1$

3. $(1-a)^2 + b^2 + c^2 =1 $

Answer 1

You don't need to approximate. You can simplify 2 and 3 together to give

$b^2 + bc + c^2 = \frac{1}{2}$

Combine that with 1 and you get an infinite set of solutions, one that has a completely defined relation between $b$ and $c$, with actually tighter limits on $b$ and $c$. $a$ is always just $1 - (b+c)$.