I have an engineering problem, which involves math. The reason it's "engineering" is that I don't need a pure mathematical solution, but a good-enough approximation could work - the only constraint is that it should be algebraically solvable.
The problem is to find a radius of a circle given three points (A,B,C) that are non-collinear outside the circle and their individual distances to the perimeter of the circle. Possible simplifying assumption is that the radius of the circle is always within the triangle formed by the three points.
The only approach I could think of is this:
Construct a system of 4 equations: three equations using the Law of Cosines for $\angle AOB$, $\angle BOC$, $\angle AOC$ ($O$ is the center of the circle), and the sum of angles $\angle AOB+\angle BOC+\angle AOC = 360^\circ$
Expand the cosines with Taylor series up to (say second degree)
Solve a system of polynomial equations.
Is there a better approach?