i redid the example and the condition that X of right point being equal to the circle origin X and being vertical impose a specific angle
meaning there is only one circle for these conditions!
Oops, my bad. That fixed x coordinate does give additional information: the angle between the radius on the x axis and a ray to the the second point, which allows a unique center to be found.
well there is this X constraint being vertical and i see it more like a specific angle then another point X and Y!
but fun to see that blender does show the real thing in 3D and that is one thing i like in blender
mostly in 3D with vectors
from mathutils.geometry import intersect_sphere_sphere_2d
from mathutils import Vector
# Point 1 (x, y, z)
start_point = Vector((-10, 0, 0))
# Point 2 (x, y, z)
end_point = Vector((0, 10, 0))
# Radius
r = 10
# Get circle center
intersect_sphere_sphere_2d(start_point[:2], r, end_point[:2], r)
# returned 2 center -> (Vector((0.0, 0.0)), Vector((-10.0, 10.0)))
# Center 1 -> Vector((0.0, 0.0))
# Center 2 -> Vector((-10.0, 10.0)