How can you find the centroid of four points ( Quadrilateral ), knowing the coordinates of these vertices?

Question

Mathematics

I would be very grateful for the formula.

Answer 1

It's well written here .
First you need to triangulate the quadrilateral. Then, the center of mass of each triangle is the arithmetic mean of the coordinates. Further, it remains to find the center of mass of two points - the centers of mass of triangles, where at each point there is a mass equal to the area of ​​the triangle.
For this to work with non-convex polygons, you need to calculate the area of ​​triangles in terms of the cross product of the sides, thus allowing negative areas for triangles outside your figure.
Final formula (in vectors):

C = ((p1+p2+p3)/3*(p1p2*p1p3)+(p3+p4+p1)/3*(p1p3*p1p4))/((p1p2*p1p3)+(p1p3*p1p4))

Here pi is the i-th vertex of the quadrilateral, pipj is the vector between points i and j. pipj*pkpl - vector product of two vectors.

Answer 2

[ (x ₁ +x ₂ +x ₃ +x ₄ )/4 ; (y ₁ +y ₂ +y ₃ +y ₄ )/4 ]