Using trigonometry, I've found that area of a regular heptagon=3.633 .

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?