Points z_1 and z_2 are two adjacent vertices of a regular n…

Points $z_1$ and $z_2$ are two adjacent vertices of a regular $n$-gon. Find the vertex $z_3$ adjacent to $z_2$ (with $z_3\neq z_1$).

Step 1 · The rotation relation

Let $\omega=e^{2\pi i/n}$. Consecutive vertices of a regular $n$-gon, going around the center $c$ in the same rotational sense as $z_1\to z_2$, satisfy:

$$ z_2-c=(z_1-c)\omega, \qquad z_3-c=(z_2-c)\omega $$

Step 2 · Solve the first relation for $c$

$$ z_2=c(1-\omega)+z_1\omega \quad\Longrightarrow\quad c=\frac{z_2-z_1\omega}{1-\omega} $$

Step 3 · Substitute into the second relation

$$ z_3=c(1-\omega)+z_2\omega=(z_2-z_1\omega)+z_2\omega=z_2(1+\omega)-z_1\omega $$

Step 4 · A cleaner equivalent form

$$ z_3-z_2=z_2\omega-z_1\omega=(z_2-z_1)\omega $$

$$ \boxed{z_3=z_2+(z_2-z_1)\,e^{2\pi i/n}} $$

Step 5 · Geometric meaning

The side vector from $z_2$ to $z_3$ is the side vector from $z_1$ to $z_2$, rotated by the exterior angle $\frac{2\pi}n$ — the defining property of a regular polygon. (The opposite orientation gives the other neighbor, $z_1$ itself, via rotation by $-\frac{2\pi}n$; the sign of the exponent selects which of the two adjacent vertices is found.)

Verification

Equal side lengths: $\ |z_3-z_2|=|(z_2-z_1)\omega|=|z_2-z_1|\,|\omega|=|z_2-z_1|\ $ since $|\omega|=1$ — confirming the new side has the same length as the given one, as required for a regular polygon.