To solve this problem, we need to find the points on the curve defined by

$x^2 - 2xy + y^2 - 4x + 8 = 0$

where the tangent lines are horizontal and vertical. We'll use implicit differentiation to get $\frac{dy}{dx}$, and then examine where it is zero (horizontal tangent) and undefined (vertical tangent).

Step 1: Differentiate Implicitly

The equation is: $x^2 - 2xy + y^2 - 4x + 8 = 0$

Differentiating both sides with respect to $x$:

1. For $x^2$, the derivative is $2x$.
2. For $-2xy$, apply the product rule: $-2 \left( y + x \frac{dy}{dx} \right)$.
3. For $y^2$, the derivative is $2y \frac{dy}{dx}$.
4. For $-4x$, the derivative is $-4$.
5. The constant $8$ has a derivative of $0$.

This gives: [ 2x - 2 \left( y + x \frac{dy}{dx} \right) +