From school days... Take positive reals x,y,z,w. The following statement
is actually symmetric in x,y,z,w: 

"there exists an equilateral triangle of side length w, 
and a point whose distances from the three vertices are x,y,z"
 
![enter image description here][1]

A quick proof: Let $ABC$ be equilateral and $P$ arbitrary. Construct $BPQ$ equilateral. 
Let $AB=AC=BC=w$, $AP=x$, $BP=y$ and $CP=z$. Then $BP=PQ=BQ=y$ by construction, $CP=z$ and $CB=x$ obviously, so it remains to check that $CQ=x$. Now note that triangle $CBQ$ is the $60^\circ$ rotation of $ABP$ around $B$.

  [1]: https://i.sstatic.net/8dMz5.png