Heptagons? I can't even draw those!

Relevant wiki: Pythagorean Theorem

The area of a heptagon is $\frac{7}{4}\times x^2 \cot(\frac{180°}{7})$ , given a side length $x$ . Call the short side of the rectangle $a$ , the long side $b$ , and the diagonal $c$ . Thus, our three equations for the areas of the heptagons are

$\frac{7}{4}\times a^2 \cot(\frac{180°}{7})$ , $\frac{7}{4}\times b^2 \cot(\frac{180°}{7})$ , $\frac{7}{4}\times c^2 \cot(\frac{180°}{7})$ .

We know due to the pythagorean theorem that $a^2 + b^2 = c^2$ . Also $k(a^2) + k(b^2) = k(c^2)$ because of the distributive property. In this case, $k = \frac{7}{4}\times \cot(\frac{180°}{7})$ . Thus, we know that $\frac{7}{4}\times a^2 \cot(\frac{180°}{7})$ = $\frac{7}{4}\times b^2 \cot(\frac{180°}{7})$ + $\frac{7}{4}\times c^2 \cot(\frac{180°}{7})$ . Then, $\boxed{R = G + B}$ .

Note: It is not necessary to know the formula for the area of a heptagon to solve this problem. All that is needed is that the formula has an $x^2$ in it somewhere, and the rest is constant.

The heptagon areas are proportional to the side lengths squared and a^2 + b^2 = c^2