Is it a coincidence that Hero's Formula defines a Super-elliptic Curve?

Question

I just realized today that Hero's formula for the area of a triangle: $$A^2=s(s-a)(s-b)(s-c)$$ satisfies an super-elliptic curve. Is there any geometric consequence to the fact that the area and the (semi)perimeter of a triangle satisfy an algebraic relation of this kind? Is this a very common occurrence in geometry that I just can't see?

Answer 1

There is a related problem known as the "Congruent number problem". It asks what are the possible areas of a rational right triangle (i.e., a right triangle with rational side lengths). A rational number is called congruent if it is the area of some rational right triangle.

It turns out that a rational number $D$ is congruent if and only if there is a point $(x,y) \in E(\mathbb{Q})$ where $y \neq 0$ and $$E : y^2 = x^3 + Dx.$$

The fact that your parameter $s$ depends on $a,b,c$ means that what you are seeing is misleading you. Well in your case you could ask whether a rational number $A$ is the area of a rational triangle. By the formula you quote this is the case if and only if there exist $a,b,c \in \mathbb{Q}$ such that $$A^2 = \frac{1}{16}(a + b + c)(a + b - c)(a - b + c)(-a + b + c)$$ where you can assume without loss of generality that $a,b,c > 0$ and $a + b > c$ (otherwise this is not a triangle).

Then we consider the semi-algebraic affine surface given by the equation $$S : (x + y + z)(x + y - z)(x - y + z)(-x + y + z) - 16A^2 = 0$$ and inequalities $x,y,z > 0$ and $x+ y> z$.

One might go out and classify this surface to try do some more in depth study. I searched for points and found that there is a point on this semi-algebraic variety when $0< A \leq 10$ (in contrast to the congruent number case).