three_pts_make_a_circle.py

from sympy import *
from sympy.solvers import solve

x = Symbol('x', real=true)
y = Symbol('y', real=true)
x0 = Symbol('x0', real=true)
x1 = Symbol('x1', real=true)
x2 = Symbol('x2', real=true)
y0 = Symbol('y0', real=true)
y1 = Symbol('y1', real=true)
y2 = Symbol('y2', real=true)

mb = (y1- y0)/(x1-x0)
mc = (y2- y1)/(x2-x1)

# midpoints
bx = (x1 + x0)/2
by = (y1 + y0)/2
cx = (x2 + x1)/2
cy = (y2 + y1)/2

# Solve system of equations to find the intersection

# equations of lines that bisect
# -1/mb (x - bx) = y - by
# -1/mc (x - cx) = y - cy

bisector1 = -1/mb *(x - bx) - y + by
bisector2 = -1/mc *(x - cx) - y + cy

# point where besectors intersect
xi = solve(bisector1 -bisector2, x)[0]
yi = solve(bisector1.subs(x, xi),y)[0]

# lengths of line segments from points to (xi,yi)
d0 = (xi-x0)**2 + (yi-y0)**2
d1 = (xi-x1)**2 + (yi-y1)**2
d2 = (xi-x2)**2 + (yi-y2)**2

# show lengths of line segments are all the same
# d0 == d1 == d2
(simplify(d1 - d0) == 0) and (simplify(d2 - d1) == 0) and (simplify(d2 - d0) == 0)