[alogo] A property of centroids of triangles

Draw to the vertices of a triangle ABC parallels and equals to sides of another triangle DEF. The resulting triangle GHI has the same centroid with the triangle ABC.

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]

Consider the sides of triangle DEF as free vectors: DE, EF, FD. Their sum is zero. Taking an arbitrary origin O, the centroid or barycenter of the triangle ABC is the point J, such that OJ = (1/3)(OA+OB+OC). Analogously the centroid of GHI would be point J', such that OJ' = (1/3)(OG+OH+OI) = (1/3)([OA+AG] + [OB+BH] + [OC+CI]) = OJ, since AG+BH+CI=0. Look at the file Centroids.html for a nice application.

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