Efficient manipulation of a list of cartesian coordinates in Python

Efficient manipulation of a list of cartesian coordinates in Python

Background:
I'm writing a program which handles large quantities of data related to
the networks of vertices of various regular shapes. I have a working
generator which produces a list of cartesian coordinates corresponding to
the vertices of said shapes based on a range of user input parameters. The
data is then passed to filters which clear up duplicate entries, sort the
data and various other functions, from where the cleaned data is fed to a
canvas module which loops through and draws the vertices.
Question:
I need to implement a new filter that efficiently loops through the
coordinates, comparing each pair against every other pair, i.e.
(x1,y1)->(x2,y2) to (x1,y1)->(xn,yn), (x2,y2)->(x3,y3) to (x2,y2)->(xn,yn)
etc. for all entries and, for example, if the relationship between (x1,y1)
and (x5,y5) fits [(x5-x1)^2+(y5-y1)^2]=vertex_spacing^2, then the two sets
of coordinates are then paired with their respective list entry numbers
and appended to a new list where one entry would be of the form: [(x1,y1),
(x5,y5), 0, 4] for example. What is the most efficient method for
achieving this?
My Attempts:
I've looked at quite a few methods for handling lists on here and on
various guides. I've attempted nested 'for' and 'if' loops, but find while
this method can work it leads to excessively long run times, as well as
attempting to breaking the problem down into numerous smaller for loops.
Further Notes:
The ultimate aim of this is to use the resulting coordinates for front-end
interface elements, and to be saved and imported as necessary. The
function of the list positions 0 and 4 in [(x1,y1), (x5,y5), 0, 4] is to
enable the interface to group coordinates for later use in canvas objects.
The method should be able to process potentially thousands of coordinates.
Thank you in advance for any help, I am of course willing to improve the
phrasing/information I've supplied and/or add example code if it is
unclear what I am asking in any way- I'm still quite new to this! :)