How to use barycentric coordinates for polygonsSuppose we want to deal with the problems with...

Step 1
It depends on your exact problem, but assuming a plane polygon, then any three non-collinear vertices of the polygon determine a triangle of reference for a barycentric coordinate system. You could use instead any other three non-collinear points in the plane of the polygon. Any points in the plane determined by the triangle, and hence the polygon vertices, will be an affine combination of the the reference triangle vertices.
Step 2
For example, if $p_1,p_2,p_3$ are the three reference points, with their existing ordinary coordinates, and p is a point in the same plane, then $p=a_1{p}_1+a_2{p}_2+a_3{p}_3$ for some real numbers such that $1=a_1+a_2+a_3$ and their values can be determined by solving a system of linear equations using the existing coordinates.