# Relation between triangle area and summed squared sidelengths

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).

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