Equiangular hexagon ABCDEF has side lengths AB = CD = EF = 1 and BC = DE = FA = r. The area of triangle ACE is 70% the area of the hexagon. What is the sum of all possible values of r?
Using Law of Cosines, I found x^2 = r^2 + r + 1. This can be found by applying the formula and using one of the 120 degrees triangles with sides 1, r, and a side of the triangle ACE (which is clearly equilateral). I'm calling the length of the unknown side x.
...Somehow or the other...I made it to 7r = r^2 + r + 1 (In my review notebook) but I am completely blanking on how in the world I got from the first thing I did to this. I probably skipped steps writing it down and now find myself in this confusedness. Would greatly appreciate someone helping me finish this solution.