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Multiplication as transformation

Multiplying a complex number by $e^{i\theta}$ rotates it by angle $\theta$ about the origin, without changing its distance from the origin. Multiplying by a positive real number $k$ scales it by $k$ . Multiplying by $k e^{i\theta}$ does both at once — this is exactly a spiral similarity, expressed algebraically.

Distances and centers

|z_1-z_2| = \text{distance between the points}, \qquad \text{midpoint} = \dfrac{z_1+z_2}{2}, \qquad \text{centroid} = \dfrac{z_1+z_2+z_3}{3}

Recognizing loci

$|z-z_0|=r$ — a circle centered at $z_0$ with radius $r$ .
$|z-z_1|=|z-z_2|$ — the perpendicular bisector of the segment joining $z_1$ and $z_2$ .

Rotating a point about another point

To rotate $z$ by angle $\theta$ about a center $z_c$ , shift to the origin, rotate, then shift back:

z' = z_c + (z-z_c)e^{i\theta}

This single formula is the workhorse for equilateral-triangle and regular-polygon construction problems in the complex plane.

Worked example

Problem: Points $A=0$ and $B=4$ are two vertices of an equilateral triangle in the complex plane. Find the two possible locations for the third vertex $C$ .

Solution: $C$ is obtained by rotating $B$ about $A$ by $\pm60°$ (the two choices give the two possible orientations of the triangle):

C = 0 + (4-0)e^{\pm i60°} = 4\left(\cos60° \pm i\sin60°\right) = 2 \pm 2i\sqrt3

So $C = 2+2i\sqrt3$ or $C=2-2i\sqrt3$ — the two reflections of each other across the real axis, as expected.

Practice problems

1.Find the midpoint of $z_1=3+2i$ and $z_2=-1+6i$ .

2.Describe the locus $|z-3|=|z+3|$ .

3.Rotate the point $z=2$ by $90°$ about the origin.

4.Find the centroid of the triangle with vertices $1+i, 3-i, -2+4i$ .

Complex Plane Geometry

Multiplication as transformation

Distances and centers

Recognizing loci

Rotating a point about another point

Practice problems