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
>technique.

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
>appropriately (???).

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:

http://www.cut-the-knot.org/triangle/TrigCeva.shtml