Single-Line Equation for Equilateral Triangle

Question

Is it possible to come up with a single-line equation in rectangular coordinates for an equilateral triangle with circumradius $R$, positioned symmetrical about the $y$-axis, as shown in the diagram below?

Answer 1

A possible solution is $$\big|y+x\sqrt3-R\big| \; +\;\big|y-x\sqrt3-R\big| \; +\;\big|2y+R\big|=3R+\delta$$ where $\delta$ is infinitesimal and $\delta>0$.

See desmos implementation here.

Each of the three terms on LHS within absolute value signs when equated to zero form the equations of the three sides of the triangle.

Interestingly, when $\delta=0$, the result is the entire region within the triangle.

Answer 2

I’m not entirely satisfied with the following, but it works and generalizes to any cyclic polygon: $$\left(\sqrt3x+y-R\right)\left(\sqrt3x-y+R\right)\left(2y+R\right)+\sqrt{R^2-x^2-y^2}=\sqrt{R^2-x^2-y^2}.$$ The idea is that you take the union of the extensions of the triangle’s sides (the term on the left) and add terms that restrict the domain (the two radicals).

Answer 3

Here's a neat solution by my friend YC:

$$\frac 2{\sqrt3}\big| x \big| \; +\; \Bigg|\frac 43 y + \frac 2{\sqrt{3}}\ \big|x\big| - \frac R3 \Bigg| \; = \; R$$

See Desmos implementation here.