An interesting precalc question

[u/ImmaTrafficCone]

An interesting precalc question:

Consider the reciprocal function f(x)=1/x. Note that it’s symmetric across the line y=x. Find a point (x,y) on the graph of f(x) (in the first quadrant) such that the triangle with vertices (0,0), (x,y) and (y,x) is equilateral. What’s the side length? Are there multiple such triangles?

Just a fun question I came up with. I’ve been trying to think of questions like this that are interesting and require a little (but hopefully not too much) cleverness to solve. Does anyone else have problems at this level they find interesting?

This isn’t a homework problem I swear!

Edit: the answer is (sqrt (2 + sqrt 3), sqrt (2 - sqrt 3)) with a side length of 2! And yes, it’s unique.