If z₁, z₂, z₃ are the vertices of an equilateral triangle ABC such that |z₁ - i| = |z₂ - i| = |z₃ - i|, then |z₁ + z₂ + z₃| equals to

$\left|{\mathrm{z}}_1-\mathrm{i}\right|=\left|{\mathrm{z}}_2-\mathrm{i}\right|=\left|{\mathrm{z}}_3-\mathrm{i}\right|$

Hence, z₁, z₂, z₃ lie on the circle whose center is i. Also given that the triangle is equilateral. Hence, centroid and circumcenter coincides.

$\begin{array}{l}\therefore \frac{{\mathrm{z}}_1+{\mathrm{z}}_2+{\mathrm{z}}_3}{3}=\mathrm{i}\\ \Rightarrow \left|{\mathrm{z}}_1+{\mathrm{z}}_2+{\mathrm{z}}_3\right|=3\end{array}$