1. ## Regular heptagons

Let $A_1A_2...........A_7, B_1B_2.........B_7, C_1C_2............C_7$ be regular heptagons with areas $S_A, S_B, S_C$ respectively. Let $A_1A_2 = B_1B_3 = C_1C_4$. Prove that $\frac {1}{2} < \frac {S_B + S_C}{S_A} < 2 - \sqrt2$.

2. Using trigonometry, I've found that area of a regular heptagon=3.633 $(side)^2$.
Let a, b, c be the side lengths of the respective sides.
I was able to prove b=0.556a or a/2<b<a.
Similarly I could express c in terms of a.
With all this data, I could find the exact value of the fraction, let alone prove the inequality.
But I think there is a more elegant way of doing this than brute force calculation.
Suggestions, anyone?