Parallel line construction without square root

Question

Given a line defined by two points having rational coordinates, is it possible to construct a parallel line at a given rational distance to the first line, using only $+$, $-$, $*$, $/$, i.e. without $\surd$?

In other words, usually one would construct a parallel line by taking the two points as a vector, normalise it, make it perpendicular, scalar multiply by the distance, add the resulting vector to one of the two original points. That normalisation employs square root, is it possible to avoid it?

Answer 1

I will assume we have a ruler with some rational numbers marked along its length.

I will also assume we have the two points on the cartesian plane, with $x$ and $y$ axes.

Let the two given points be $(x_1,y_1)$ and $(x_2,y_2)$, and let the required distance between the parallel lines be $d$. Here is the procedure:

1. On the $x$ axis, mark points $A(y_1-y_2 , 0)$ and $B(y_2-y_1 , 0)$
2. On the $y$ axis, mark points $C(0 , x_1-x_2)$ and $D(0 , x_2-x_1)$
3. Draw $AC$ and $BD$ (and extend them if necessary) and let them meet the (if necessary, extended) initially given line at $E$ and $F$, respectively.
4. Using the ruler, mark points $G$ and $H$ on extensions of $AE$ and $BF$ at distance $d$ from $E$ and $F$, respectively.
5. Draw $GH$. It is the required line.

The following figure visualizes this procedure. The two given points are marked with + signs.