Minimize $\;\left(-x+y+1 \right)^2 + \left( x-y-2\right)^2 + \left(x+2y-3 \right)^2 \;$ without using partial derivatives

By completing the square, $$ \big(-x+y+1 \big)^2 + \big( x-y-2\big)^2 + \big(x+2y-3 \big)^2 = 3(x-2)^2+6(y-\frac{1}{2})^2 +\frac{1}{2}$$ the minimun is $\frac{1}{2}$

hint:

$a = y-x+1, b = x-y-2, c = x+2y-3$, then find an equation in $a,b,c$ and use CS inequality.