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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).

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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

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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.

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  • $\begingroup$ I am not looking for reformulations of Heron's formula. This introduces additional triangle side lengths or products or side lengths as another parameter. Is there anything interesting just looking at the ratio of the triangle and square areas? $\endgroup$ – granular bastard Jan 13 at 13:54
  • $\begingroup$ You should edit your question to reflect such restrictions. $\endgroup$ – an4s Jan 14 at 9:08
  • $\begingroup$ I have edited the original question. $\endgroup$ – granular bastard Jan 14 at 13:29

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