Given two 2D line segments, $\overline{ab}$ and $\overline{cd}$, and a point $p$, I would like to find a scalar value $t$ such that the line segment between $\overline{ab}(t)$ and $\overline{cd}(t)$ is at a distance of exactly $L$ from $p$.
Clarifications:
$\overline{ab}(t)$ is the point $a * (1 - t) + b * t$
The points $a$, $b$, $c$, $d$, and $p$ are known. The distance $L$ is known. Only the scalar value $t$ is unknown.
Since $\frac{(-ab(t)+p).(cd(t)-ab(t))}{(cd(t)-ab(t))^2}(cd(t)-ab(t))$ is the projection $-ab(t)+p$ onto $cd(t)-ab(t)$, $$L^2=(-ab(t)+p)^2-\left(\frac{(-ab(t)+p).(cd(t)-ab(t))}{(cd(t)-ab(t))^2}(cd(t)-ab(t))\right)^2$$ I believe it's the equation for $t$ you're looking for :)