Simple proofs: The impossibility of trisection

The article explains the famous ancient Greek problem of trisecting an arbitrary angle using only a compass and an unmarked straightedge and why it is impossible. It presents an elementary proof based on algebra rather than advanced Galois theory: the key idea is that ruler-and-compass constructions can only create numbers whose algebraic degrees are powers of 2, while trisecting a 60° angle would require constructing $(\cos(20^\circ))$, which satisfies a cubic equation of degree 3. Since a degree-3 number cannot come from the allowed ruler-and-compass operations, an arbitrary angle trisection is impossible. The article also connects the problem with the other classical construction problems—squaring the circle and doubling the cube—and shows how geometry, trigonometry, and algebra combine to prove the limitation of ancient construction methods.