I know that in standard two-dimensional Euclidean space three non-collinear points have a unique circle that touches all three points. I saw that @dan uznanski gave a determinant form for finding the equation of of the circle using minor determinants of a specific matrix. I also saw a development similar to this in Felix Klein's elementry mathematics from an advance viewpoint book.
I was curious about circles in the taxicab geometry. I'm trying to make an analogue of the prime gap bound for Gaussian integers. I'm defining the prime gap through the biggest possible circle between primes in a taxicab geometry.
This led me to question a few things about taxicab cirlces. Notice that the points (-1,1), (0,0), and (1,1) fit on the circles |x-0|+|y-b|=b for b>1. These points are not collinear in a standard sense, but the point (0,0) falling on a corner of this taxicab circle makes them collinear. Is there a simple generalization of what it means to be non-collinear in a taxicab geometry so that we can have a unique taxicab circle fit through three non-collinear points? What are the general formulas for the center and radius of a taxicab circle that falls through three points that are not collinear under this more general definition of "collinear"? Is there a generalized concept for the determinant description for standard Euclidean circle centers and radius in context of the taxicab geometry? I'd love to hear your comments.