Relation between triangle area and summed squared sidelengths

Question

I am wondering if there are any interesting relations or interpretations between the total area of the squares and the triangle area?

I am not looking for variants of Heron's formula. I am looking for a relation in the style: "the ratio between the total square area and triangle area is proportional to the circumradius of the triangle" (fictive sentence).

Answer 1

Yes, there is. Let the squares have areas $A,B,C$ then the area $T$ of the triangle satisfies $$16T^2=4(AB+BC+CA)-(A+B+C)^2.$$

See https://en.wikipedia.org/wiki/Heron%27s_formula

Answer 2

Let the area of each square be $A_1, A_2, A_3$. The three sides of the triangle are then $\sqrt{A_1}, \sqrt{A_2}, \sqrt{A_3}$, respectively. The perimeter of the triangle is $\sqrt{A_1} + \sqrt{A_2} + \sqrt{A_3}$. Let $p$ be half of the perimeter of the triangle, i.e., $$p = \frac{\sqrt{A_1} + \sqrt{A_2} + \sqrt{A_3}}2.$$ Then, from Heron's formula, we have $$A_T = \sqrt{p\left(p - \sqrt{A_1}\right)\left(p - \sqrt{A_2}\right)\left(p - \sqrt{A_3}\right)},$$ where $A_T$ is the area of the triangle.