Suppose that we are given 40 points equally spaced around the perimeter of a square, so that four of them are located at the vertices and the remaining points divide each side into ten congruent segments. If P, Q, and R are chosen to be any three of these points which are not collinear, then how many different possible positions are there for the centroid of triangle PQR?
Number of distinct centroids of triangles formed by 40 equally spaced points on a the perimeter of a square
3
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combinatorics
geometry
triangles
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1I think that if you take coordinates so your square is $[0,10]\times[0,10]$ and use that the centroid of $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$ is $\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right),$then everything reduces to a combinatorics problem. My conjecture is that all points $(x,y)\in(0,10)\times(0,10)$ such that $3x,3y\in\mathbb{Z}$ are centroids of some triangle. – 2012-11-02