Let's say you have two points, $(x_0, y_0)$ and $(x_1, y_1)$.
The gradient of the line between them is:
$$m = (y_1 - y_0)/(x_1 - x_0)$$
And therefore the equation of the line between them is:
$$y = m (x - x_0) + y_0$$
Now, since I want another point along this line, but a distance $d$ away from $(x_0, y_0)$, I will get an equation of a circle with radius $d$ with a center $(x_0, y_0)$ then find the point of intersection between the circle equation and the line equation.
Circle Equation w/ radius $d$:
$$(x - x_0)^2 + (y - y_0)^2 = d^2$$
Now, if I replace $y$ in the circle equation with $m(x - x_0) + y_0$ I get:
$$(x - x_0)^2 + m^2(x - x_0)^2 = d^2$$
I factor is out and simplify it and I get:
$$x = x_0 \pm d/ \sqrt{1 + m^2}$$
However, upon testing this equation out it seems that it does not work! Is there an obvious error that I have made in my theoretical side or have I just been fluffing up my calculations?