Given any line segment, finding the point P(x, y) that divides the segment AB into a specific ratio r can be done by calculating the following:
$$x = \frac{x_1+x_2r}{1+r}$$ $$y = \frac{y_1+y_2r}{1+r}$$
Now I'm have the current problem: given a line segment AB which is equally divided into five segments, calculate the ratio of every point that divides AB.
What I have so far is that there most be four points $P_1$, $P_2$, $P_3$, $P_4$ in order to divide the segment, and their ratios are given by:
$$r_1=\frac{AP_1}{P_1B}$$ $$r_2=\frac{AP_2}{P_2B}$$ $$r_3=\frac{AP_3}{P_3B}$$ $$r_4=\frac{AP_4}{P_4B}$$
But that's as far as I've got, I'm stuck here. How can I calculate those ratios? Any thoughts?
1) Find $P_4$ which divides AB in the ratio 4 : 1
2) Note that $P_2$ is the midpoint of $AP_4$.
3) $P_1$ is ….
4) $P_3$ is ....