It's a really cool geometric interpretation for barycentric coordinates, but does an area-based viewpoint have any useful applications?
How can we extend this to higher dimensions?
@cooliotonyio it's useful for drawing the connection to discussing the actual barycenter of the triangle (i.e. its center of mass). technically you can compute the barycentric coordinates this way as well.
@Hexhu i believe that barycentric coordinates also have a similar interpretation in 3D - for example, wikipedia has an section dedicated to barycentric coordinates in tetrahedra