# Equidistant Parallel Lines

*Property discovered by Ada while researching for new paper folding patterns.*

There are 5 equidistant parallel lines A_{0}B_{0}, A_{1}B_{1}, A_{2}B_{2}, A_{3}B_{3} and A_{4}B_{4}.

The line CD intersects A_{2}B_{2}, A_{3}B_{3} and A_{4}B_{4} in F_{2}, F_{3} and F_{4}.

The line CE intersects A_{1}B_{1}, A_{2}B_{2} and A_{3}B_{3} in G_{1}, G_{2} and G_{3}.

F_{2}G_{1} intersects A_{0}B_{0} in H_{0}.

F_{3}G_{2} intersects A_{1}B_{1} in H_{1}.

F_{4}G_{3} intersects A_{2}B_{2} in H_{2}.

Then H_{0}, H_{1} and H_{2} are collinear.

Proof:

We first notice that:

Because this are alternative routes from to we have:

We add the last two equalities and get:

But the first and third from the right side are 0:

We multiply the last equality by 2 and get:

We also notice that:

So if we do the replacements we get:

(1)

Because this are alternative routes from to we have:

We add the last two equalities and get:

but the first from the right is zero

because of (1) we have:

so are collinear

Note:

This property leads to the following recursive pattern: