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"
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 $CB=w$ obviously, so it remains to check that $CQ=x$. Now note that triangle $CBQ$ is the $60^\circ$ rotation of $ABP$ around $B$.
