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1988 OIM Problems/Problem 4

Problem

Let $ABC$ be a triangle which sides are $a$, $b$, $c$. We divide each side of the triangle in $n$ equal segments. Let $S$ be the sum of the squares of the distances from each vertex to each of the points dividing the opposite side different from the vertices.

Prove that $\frac{S}{a^2+b^2+c^2}$ is rational.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

https://www.oma.org.ar/enunciados/ibe3.htm

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