Find the point of intersection of the asymptotes of the graph of
\[y = \frac{x^2 - 4x + 3}{x^2 - 4x + 4}.\]
The denominator factors as $x^2 - 4x + 4 = (x - 2)^2,$ so the vertical asymptote is $x = 2.$

Since
\[y = \frac{x^2 - 4x + 3}{x^2 - 4x + 4} = \frac{(x^2 - 4x + 4) - 1}{x^2 - 4x + 4} = 1 - \frac{1}{x^2 - 4x + 4}.\]Thus, the horizontal asymptote is $y = 1,$ and the intersection of the two asymptote is $\boxed{(2,1)}.$