I am not sure I understand the problem. I'll ask a couple of questions:
>Question: I will give you a brief scope of the problem.
>Suppose A, B and C are some devices which have GPS device so
>they know their exact locations (xi's,yi's) and we wish to
>determine the location of P. If I know the locations of A, B
>and C and the six inter-node distances, I can solve for xp
>and yp using a linear equation.
What are the six inter-node distances, or the nodes themselves, for that matter?
>I need the distances to compute areas (and, in turn
>barycenters). Is there any other technique to compute these
>barycenters, for instance, using only the angles in the four
>triangles generated by A,B,C,P. or some other weighting
Are you assuming that Earth is flat? Given that you mention the GPS, ignoring the curvature appears, say, light hearted.
>I know area formula using radius of circumscribed circle and
>sine of angles but that won't help coz of the radius. My
>hope is that since barycenters are ratios
But sine is also a ratio.
>... they should be
>indifferent to distances, or radii for that matter. coz the
>barycenters wont change if I scale the coordinates
This is why I asked about the curvature. Scaling is less meaningful on a sphere than on a plane.
>so there must be some representation
>with orientation or angles ?
There is a trigonometric form of Ceva's theorem that might be relavant to your inquiry: