*All measures are in degrees. Since $\text{gcd}(n,3)=1$, there exist $x,y \in \mathbb Z$ such that $3x+ny=1$. Multiplying both sides by $\frac{60}{n}$ gives $(3x+ny)\frac{60}{n}=\frac{60}{n}=\frac{180}{3n}$ $\frac{180x}{n}+60y=\left(\frac{180}{n}\right)\cdot x + 60 \cdot y \frac{180}{3n}$. Since $\frac{180}{n}$ and $60$ can be constructed, and $x$ and $y$ are integers, we can copy an angle of measure $\frac{180}{n}$ $x$ times, and copy a $60^{\circ}$ angle $y$ times. Combining the angles gives an angle of measure $\frac{180}{3n}$, which is $\frac{180}{n}$ trisected.

Since $\frac{\phi(3n)}{\phi(n)}= 2$, it follows that $\mathbb{Q}(\zeta_{3n})$ is a quadratic extension of $\mathbb{Q}(\zeta_n)$.