Originally Posted by

**CaptainBlack** (minor variant of a problem due to Roy Barbara)

Let $\displaystyle a,\ b,\ c$ be three positive real numbers.

Find necessary and sufficient conditions on $\displaystyle a,\ b,\ c$ for there to exist an interior point $\displaystyle P$ in the equilateral triangle $\displaystyle ABC$ with unit side, such that $\displaystyle |PA|=a,\ |PB|=b,\ |PC|=c$.

*I have a solution to this, but as I have not looked it up so I cannot tell if it is the originator's solution but as it is not as neat as I would like it is probably clumsy compared to the best solution, so lets see what we can do*

CB