finding the coordinates of a point given a distance

Question

I have a fixed point $(a,b)$ and movable point $(c,d)$. The distance between these two points is $r$. Now, I want to increase the distance to, say, $r'$. What are the new coordinates of the new point $(c',d')$ with respect to the original so that $(a,b), (c,d), (c',d')$ are collinear.

Answer 1

The triplets of points $\{(a,b),(c,d),(c,a)\}$ makes a right triangle where the segment $(a,b)-(c,d)$ is the hypothenuse and has length $r$. Assuming the new point at distance $r'$ moves in the same direction as the hypothenusa, you obtain a new similar right triangle and the length of the new cathetus can be trivially found by similarity.

$$\left\{\begin{align} \frac{c'-a}{c-a}&=\frac{r'}{r} \\ \frac{d'-b}{d-b}&=\frac{r'}{r} \end{align}\right. $$