The problem is from Kiselev's Geometry exercise 484:
Divide a given angle congruent to 1/7th of the full angle into: (1) three congruent parts; (2) five congruent parts.
What I know is that, as listed in the book, 'one can divide a circle only into such a composite number of congruent parts which contains no other factors except: (1) prime factors of the form $2^{2^n} + 1$ in the first power; (2) the factor $2$ in any power.' Therefore, the given angle cannot be constructed with a straightedge and compass from a circle. I also found that there exists no general method to trisect an angle.
The only problem that has even a bit of resemblance to this question is exercise 476:
Prove that if $ABCDEFG$ is a regular $7$-gon, then $1/AB = 1/AC + 1/AD$.
It is solved by the law of sines and some trigonometric formulas. I am not sure whether the same technique is used in solving the first problem.
I do not know how to approach this problem. The problem is from the chapter of regular polygons. I could not think of any reason why this specific angle is chosen and why this angle admits trisection and quintisection, let alone how to construct it. Any help would be greatly appreciated.