How can we draw a line between two distant points using a finite-length ruler?

We want to join by a line two distinct points $A$ and $B$. We have only a ruler of length $l>0$ and a pen. If $AB>l$ how can we do this?

Solution 1:

This is one instance of Desargues' theorem:

Use the ruler to draw two arbitrary lines through $A$ in the general direction of $B$. You can extend them by simply shifting the ruler along the portion of the line you've already drawn. Once you're close to $B$, end the lines in points $C$ and $D$. On the line $CD$ choose a third point $E$, and through $E$ choose two more lines, one which intersects $AC$ in $F$ and $AD$ in $G$, the other which intersects $BC$ in $I$ and $BD$ in $H$. Now $GH$ and $FI$ intersect in a point on $AB$. If you have made the arbitrary choices in a suitable fashion, then all those lines could be drawn with your short ruler, and the resulting point $P$ will be closer to $A$ than $B$ was. Externd line $BP$ and it will reach $A$.

If any of the lines needed for this construction is too long, you may iteratively perform this whole construction there, or pick your points such that they lie closer together.

As an alternative, you might use Pappos' theorem instead of Desargues'. It requires one point and one line less, and on the whole looks somewhat more symmetric:

As before, you start by drawing lines $AC$ and $AD$ in the general direction of $B$. You choose a point $E$ on $BC$ and a point $F$ on $BD$. Then you intersect $FC$ with $AD$ to obtain $G$ and $ED$ with $AC$ to obtain $H$. $EG$ and $FH$ will intersect in $P$ which lies on $AB$. Externd $BP$ and it reaches $A$, as before.