Exercise 421Algebra › Complex numbersHigh schoolAdded 3 Nov 2026

Midpoint and centroid in the complex plane

#midpoint#centroid#complex plane

Statement

Let $A$ , $B$ , $C$ have affixes $z_A,z_B,z_C$ . Show that:
- the midpoint of $AB$ has affix $\frac{z_A+z_B}{2}$ ;
- the centroid of $ABC$ has affix $\frac{z_A+z_B+z_C}{3}$ .

Hint

Express the coordinates.

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Solution

Midpoint. If $z_A=a+bi$ and $z_B=c+di$ , the midpoint has coordinates $((a+c)/2,(b+d)/2)$ , affix $\frac{a+c}{2}+i\frac{b+d}{2}=\frac{z_A+z_B}{2}$ .

Centroid. Similarly, the centroid has coordinates the average of the three coordinates, giving affix $\frac{z_A+z_B+z_C}{3}$ .

Conclusion. Both formulas hold; they are the natural complex analogues of vector averaging.

Midpoint and centroid in the complex plane

Statement

Solution

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