Prove that for some positive integer \(N\), \(N\) points can be chosen on a circle such that there are at least \(1000N^2\) unordered quadruples \((A,B,C,D)\) of distinct selected points for which \(\displaystyle \frac{AC}{BC} = \frac{AD}{BD}\).
Problem
Source: XIII International Festival of Young Mathematicians Sozopol 2024, Theme for 10-12 grade
Tags: algebra, combinatorics, geometry, number theory