Travelling salesman using brute-force and heuristics
I enjoyed the first look at the code as it's very clean, you have
extensive docstrings and great, expressive function names.  Now you know
the deal with PEP8, but except for the one 200 character long line I
don't think it matters much really.

There're a few typo with the wrong spelling "algoritmh".

The coordinates should be immutable 2-tuples.  The reason being the
safety of immutable data-structures.  YMMV, but that makes it really
obvious that those are coordinates as well.

optimized_travelling_salesman should make a defensive copy of
points, or you should otherwise indicate that it's destructive on that
argument.

Instead of if start is None: start = points[0] you could also use
start = start or points[0] to save some space while still being
relatively readable.

For the algorithms the only thing I'd is not to use square root if you
don't have to.  You can basically create a distance_squared and use that
instead of distance because the relationship
between a smaller and bigger distance will stay the same regardless.
That doesn't apply for the final output of course. Edit: And, as mentioned below by @JanneKarila, you can't use that for the brute-force version.