Determine all triangles ABC such that all of \( \frac{AB}{BC}, \frac{BC}{CA}, \frac{CA}{AB}, \frac{\angle A}{\angle B}, \frac{\angle B}{\angle C}, \frac{\angle C}{\angle A}\) are rational.

The best solution was submitted by 황성호. Congratulations!

Alternative solutions were submitted by 정성진(+3), 이영민(+3), 채석주(+3), 이종원(+3), 장기정(+3), 배형진(+3), 남재현(+2), 김경민(+2), 박경호(+2), 서웅찬(+2). Thank you for your participation.