My uncle once argued a case in the U.S. Supreme Court regarding Indiana & Kentucky's border, which was initially established in the middle of the river.

The Ohio River has gradually shifted south since the border was established, so some parts of Kentucky are now on the "wrong" side of the river.

Wondering the same, don't enlightenment there would be great

As an example, here is what is said by Oregon & Washington about the border between them. This was ratified by both states in a compact. It looks like they switched from less precise definitions like "the varying center of the main channel of the Columbia River" to a list of precise coordinates.

Washington: https://app.leg.wa.gov/rcw/default.aspx?cite=43.58&full=true

Oregon: https://oregon.public.law/statutes/ors_186.520

Lawyers don't really think about these things in the same way that mathematicians do:

https://en.wikipedia.org/wiki/Thalweg#Thalweg_principle

> The Treaty of Versailles, for example, specifies that "In the case of boundaries which are defined by a navigable waterway" the boundary is to follow "the median line of the principal channel of navigation."

I've thought about this before and came to the non-rigorous conclusion that if you define the middle as always being equidistant between the nearest points on the shoreline, then its length must be finite. The same is true if you define it as the shortest path through the river.

Basically this is because an infinite-length line must "loop back" on itself, so that at some point it curves by at least 90° from the main direction of movement. This can never happen at a small scale for the middle of a river.

I would really love a mathematician to weigh in on this though. The one main problem I can see is that it might lead to something akin to the Weierstrass function https://en.wikipedia.org/wiki/Weierstrass_function which itself has infinite length, although I don't quite understand why this is or if it's applicable to the case of rivers.