The circumcenter of mass is a concept that can be defined by analogy with one of the the constructions for the geometric centroid
for the case of polygons. The geometric centroid
of a polygon may be determined by triangulating
the polygon using any interior point, computing the center of mass of each triangle,
taking the weighted sum over triangles where weights are taken as individual triangle
areas, and dividing the result by the total area of the polygon. Doing the same procedure
but taking the circumcenter instead of the geometric
centroid for each triangle gives the circumcenter of mass, whose value turns
out to be independent of how the original polygon is triangulated.

This construction is mentioned in Laisant (1887, pp. 150-151), who attribute it to the Italian algebraic geometer G. Bellavitis (Tabachnikov and Tsukerma 2015).

Explicit formulas of the circumcenter of mass can be given by

(1)

(2)

(Tabachnikov and Tsukerma 2015) or

(3)

(4)

(Tabachnikov and Tsukerma 2014) for a polygon with vertices , ..., , area , and where indices outside the range are taken to refer to cyclically repeating vertices.

Adler, V. "Cutting of Polygons." Funct. Anal. Appl.27, 141-143, 1993.Laisant, C.-A. Théorie
et applications des équipollences. Paris: Gauthier-Villars, pp. 150-151,
1887.Tabachnikov, S. and Tsukerman, E. "Circumcenter of Mass and
Generalized Euler Line." Disc. Comput. Geom.51, 815-836, 2014.Tabachnikov,
S. and Tsukerman, E. "Remarks on the Circumcenter of Mass." Arnold Math.
J.1, 101-112, 2015.