<s>[INST] A question about an equilateral triangle Suppose that $\triangle ABC$ is an equilateral triangle. Let $D$ be a point inside the triangle so that $\overline{DA}=13$, $\overline{DB}=12$, and $\overline{DC}=5$. Find the length of $\overline{AB}$.
 [/INST] The Wikipedia article on equilateral triangles quotes the following theorem from "Curious Properties of the Circumcircle and Incircle of an Equilateral Triangle," by Prithwijit De (http://ms.appliedprobability.org/data/files/Abstracts%2041/41-1-7.pdf):
Theorem: Let $ABC$ be an equilateral triangle with side $s$, and let $P$ be a point in the plane of the triangle with distances $p$, $q$, and $r$ to $A$, $B$, and $C$, respectively. Then $$3(p^4+q^4+r^4+s^4) = (p^2+q^2+r^2+s^2)^2{\textrm.}$$
Solving $3(5^4+12^4+13^4+s^4) = (5^2+12^2+13^2+s^2)^2$ for $s$ gives $s=\sqrt{169\pm 60\sqrt3}$. Because $P$ is inside the triangle, $s>13$, which allows only $s=\sqrt{169+ 60\sqrt3}$, the same solution others have obtained.
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