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.