TITLE: Position of a point with fixed distance between other two points
QUESTION [5 upvotes]: I have two points, $p_1$ and $p_2$, in a cartesian plane, and a fixed radius, $r$.
I want to find the coordinates of another point, $p_3$, that is in the same line of the $p_1$ and $p_2$, and always in a fixed distance, $r$, from the point $p_1$.
$(a,b)$       $(?,?)$                      $\quad \quad \quad\space(c,d)$ 
$p_1$---------$p_3$---------------------$p_2$
        $\quad r$  
Considering that the points $p_1$ and $p_2$ can be on anywhere in the plane.
Thanks in advance.

REPLY [2 votes]: Let's define point $P_3$ as $P_3(e,f)$ . 
You may calculate coordinates $e$ and $f$ using following equalities :
$e=\frac{a+\lambda c}{1+\lambda}$ , and $f= \frac{b+\lambda d}{1+\lambda}$ 
where $\lambda=\frac{r}{|P_1P_2|-r}$ , and $|P_1P_2|=\sqrt{(a-c)^2+(b-d)^2}$