I'm having trouble understanding the motivation between finding the centroid of a quadrilateral.
Q: Find the centroid of a quadrilateral with vertices at (-8,12), (7,15) (13,-9), and (-2,-3).
I've solved the problem, using the procedure described @How can I construct the centroid of a quadrilateral? (by finding the intersection of the 4 centroids of the triangles formed by the 2 diagonals of a quadrilateral). However, I've been struggling to understand why this method gives the centroid of a quadrilateral (perhaps due to a theorem?).
** To clarify, I found the centroid which divides the quadrilateral into 4 equal areas
Best Answer
The key property, which can be derived by the definition, is that the centroid of a system of $2$ objects lies on the line which connects the centroid of each single object.
Then dividing the quadrilateral by a diagonal we find a first segment that contains the centroid of the quadrilateral and dividing by other diagonal we find a second segment that contains the centroid of the quadrilateral. Therefore the centroid coincides with the intersection of the two segments.