1985 OIM Problems/Problem 2

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Problem

Let $P$ be a point in the interior of equilateral triangle $ABC$ such that: \[PA=5,\;PB=7,\; and \; PC=8\] Find the length of one side of the triangle $ABC$

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

Solution

By Viviani's Theorem, the altitude of the triangle is the sum of the given lengths, or $20$. It follows that the side length is $\boxed{\frac{40\sqrt3}{3}}$.

See also

https://www.oma.org.ar/enunciados/ibe1.htm