Here are the steps to divide a line segment into $n$ equal segments using only the tools in affine geometry. Here, we assume the axiom that there is a unique line through a given point that is parallel to a given line, i.e. the two lines do not meet.

Construct the line segment $\overline{AB}$, given points $A$ and $B$ in affine $n$-space.

Take one of the points on the line segment, say $A$, and draw a line through $A$ not parallel to the line $AB$.

Pick a point $C_1$ on the line constructed in Step 2.

Draw a line through $A$ not parallel to the line $AB$ or $AC_1$, and pick a point $D_1$ on this line.

Construct a parallel line to the line $AC_1$ through the point $D_1$.

Construct a parallel line to the line $AD_1$ through the point $C_1$, and define the meet (intersection) of this line with the line in Step 4 to be the point $D_2$.

Draw the line $D_1C_1$ and construct a parallel line to this line through the point $D_2$.

Define $C_2$ to be the meet of the line constructed in Step 6 with the line $AC_1$.

Repeat Steps 5-7, now with $C_1$, $C_2$ and $D_2$ instead of $A$, $C_1$ and $D_1$; we do this $n-1$ times, so that we get $n$ copies of the line segment $\overline{AC_1}$.

Suppose then that the last point we have is $C_n$; we then draw the line $BC_n$.

Through each of the points $C_i$, for $i$ ranging from $1$ to $n-1$, draw the lines through each of the points, parallel to the line drawn in Step 9.

The meets of each of the lines in Step 10 and the line segment $\overline{AB}$ will then be denoted by $E_i$, for $i$ ranging from $1$ to $n-1$.

As a result of this construction, we then have that the points $E_i$, for $i$ ranging from $1$ to $n-1$, divide the line segment $\overline{AB}$ into $n$ equal line segments.

I did a construction (see picture above) where I just divided a simple line segment into $3$ equal segments.

EDIT: Picture too small to look at by eye; not sure how to make it bigger. Just click to enlarge, I guess.

EDIT 2: There are a lot more things to say about this, but I am going to write a book with this content in it. Don't want to spoil too much...

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