Proving a 9-point circle Given the equation for a circle and 2-9 points on the circumference of the circle, I am trying to prove whether the points on the circle are part of a possible non-obtuse 9-point circle.
I have no idea where to start.
From what I can tell no one has tried this yet, at least I can't find anything on it. But this is way over my head, so perhaps the answer was just staring me in the face.
EDIT: Well, the bot just told me to make this clearer, so here we go.
I have a user input circle, which I've simplified to an equation (for the sake of example let's say it's $(x-0)^2+(y-0)^2=1^2$, which gives us a diameter $2$ circle originating from $(0,0)$), and two (2-9 really, depends on the case for the sake of the problem 2) points which fall on the circumference
(let's say $(0,1)$ and $(0.64, 0.768)$) I need a way to test (with python) whether or not the two points are part of a possible nine-point circle.
 A: There are two ways to read your question. One of them is boring:

Given up to 9 points on a circle, is there a non-obtuse triangle such that all the given points lie on the 9-point circle of that triangle?

Or in other words, can that circle be a 9-point circle? The answer to this is trivially yes. Just construct any equilateral triangle circumscribed around that circle. Since the incircle of an equilateral triangle is also its 9-point circle, you have found a suitable triangle.
The more interesting question asks whether your inputs are among the specific points for which the 9-point circle was named:

Given up to 9 points on a circle, is there a non-obtuse triangle such that each of the given points is either a side midpoint or an altitude foot or the midpoint between orthocenter and a vertex of that triangle?

The first question here is which of the given points corresponds to which of the points on the circle. There are $9!=362{,}880$ possible permutations if you have $9$ given points. But actually in most situations $3$ of the inputs would be enough. So you have $\binom n3$ ways to pick $3$ points from your $n$ inputs (with $3\le n\le 9$, ignoring the case of $n=2$ inputs which is too boring), and $\frac{9!}{(9-3)!}=504$ ways how these three points can map to 3 specific points of the 9-point circle.
So iterate over all possible mappings, then try to reconstruct the triangle vertices from the given points, check whether it would be obtuse, and also check whether all the other points correspond to some of the 9-point circle points of that triangle.
How would you reconstruct the triangle? Maybe there are smarter ways for doing this, but you can assume two variables as the coordinates of each vertex. You can then compute the 9 points and have rational functions for their coordinates. From your 3 selected input points you get 6 coordinates, and by equating these numbers to the rational functions you end up with six equations of the form $p/q=r$, where $p,q\in\mathbb Q[A_x,A_y,B_x,B_y,C_x,C_y]$ are polynomials in the unknowns and $r\in\mathbb R$ is your input coordinate. Reformulate this as $p-rq=0$ then use standard techniques for solving non-linear multi-variate systems of polynomial equations, e.g. repeated application of resultants and the likes.
Or get a computer algebra system to do this for you. You mention Python, so I would probably attempt to pull in Sage, formulate the above as an ideal of hopefully dimension zero, then compute its variety to obtain the actual points. The system will probably use Gröbner bases behind the scenes to manipulate the ideal representation, and I'll be happy if I don't have to implement that myself.
You might want to explore barycentric coordinates or trilinear coordinates as potential alternatives. I think for those you would only need the triangle edge lengths as variables, so you would have 3 variables not 6. On the other hand, it is less clear how to obtain 3 equations for 6 input coordinates in this setup, and also computing the actual triangle corners from the result might be harder.
Note that numeric precision will probably make things problematic. Deciding whether a point lies on a circle usually corresponds to some expression being zero, but with numeric errors you may need to treat almost zero as good enough. Deciding the threshold will be tricky. You might also find that for some of the choices of 3 points to reconstruct the triangle, that reconstruction will be very poorly conditioned numerically. Small amounts in input errors would translate to vastly different triangles.
For these reasons, you might want to avoid selecting 3 points to reconstruct the triangle, and instead take all 9 points into consideration and find the best matching triangle for all of them. At this stage you would have a non-linear optimization problem, and again I'd recommend you use some pre-made tools for that because implementing an optimizer for those by hand is going to be very painful.
