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:

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