How do I find the middle(1/2), 1/3, 1/4, etc, of a line?

Think about the formula for the midpoint of a line as $$x = \frac{1}{2}x_{1} + \frac{1}{2}x_{2}.$$ We find the midpoint by taking equal contributions from either end of the line.

If we want to find a point which is one third of the way along the line we should use the following $$x = \frac{2}{3}x_{1} + \frac{1}{3}x_{2}$$ where we take uneven contributions from the two endpoints.

The point one quarter of the way along the line can be found using $$x = \frac{3}{4}x_{1} + \frac{1}{4}x_{2}.$$

The y-coordinates are found in a similar manner.

Divide your segment into even parts.

From the last point you can find that $$ x_2 = x_1 + n\Delta \implies \Delta = \frac {x_2 - x_1}n $$ You need to find second point in that division, so $$ x' = x_1 + \Delta = x_1 + \frac {x_2-x_1}n = \frac {n-1}n x_1 + \frac 1n x_2 $$ So, let's say you want to divide your segment into 2 even parts, then $n = 2$ and $$ x' = \frac {x_1}2 + \frac {x_2}2 $$ If $n = 3$ (3 even parts) $$ x' = \frac 23 x_1 + \frac 13 x_2 $$ $n = 4$ (4 even parts) $$ x' = \frac 34 x_1 + \frac 14 x_2 $$