(Originais Teachy 2023) - Question Hard of Mathematics

Consider a triangle ABC in the Cartesian plane, where A = (2,3), B = (4,1), and C = (6,5). An Architecture student is designing a new building and wants to position a support pillar at the centroid of the triangle, which is the point of intersection of the medians. A sensor spring will be installed at the top of the pillar to measure the force exerted by the structure. The force measured by the spring is directly proportional to the sum of the squares of the distances from the pillar to the vertices. Calculate the coordinates of the centroid of the triangle and then calculate the sum of the squares of the distances from the centroid to the vertices of the triangle. Based on this information, explain how the centroid can be a strategic location for the installation of the pillar, considering the principles of balance and force distribution in architectural structures.