Geometry with complex numbers

Summary

This lesson uses complex numbers as coordinates in the plane and translates the notions of elementary geometry into algebraic operations: distance is the modulus of a difference, the midpoint and the centroid are averages, and the motions of the plane are short formulas. It proves that $R(z) = c + e^{i\varphi}(z - c)$ is the rotation about $c$ through the angle $\varphi$ , that the reflection in a line through the origin with angle $\alpha$ is $S(z) = e^{2i\alpha}\conj{z}$ and that the composition of two reflections is a rotation. It characterizes collinearity and perpendicularity by means of a quotient, defines the orientation of a triangle by the sign of the imaginary part of that quotient, proves that three points form an equilateral triangle if and only if $z_1 + \omega z_2 + \omega^2 z_3 = 0$ , with $\omega = e^{\frac{2\pi i}{3}}$ , and shows that the points whose distances to two fixed points are in a constant ratio form a circle, the circle of Apollonius. The problems culminate in Napoleon’s theorem.

Learning objectives

Represent points, distances, midpoints and centroids with complex numbers, and express translations, rotations, dilations and reflections as functions of a complex variable.
Prove the formulas for the rotation about a point and for the reflection in a line, and study the compositions of reflections and of rotations.
Decide collinearity, perpendicularity and the angle between two line segments by means of the quotient of two differences, determine the orientation of a triangle by the sign of the imaginary part of that quotient, and characterize the equilateral triangle with the cube root of unity $\omega$ .
Describe circles with the modulus, prove the theorem on the circle of Apollonius and use these tools in classical problems, such as those of Napoleon and of Van Aubel.

Points, distances and centers

Since the Algebra and Geometry I lesson on operations with complex numbers, each complex number $z = a + bi$ is identified with the point $(a, b)$ of the plane, and the difference $z - w$ is the vector that goes from the point $w$ to the point $z$ ; by the Pythagorean theorem, the distance between the two points is $\abs{z - w}$ . In this lesson the point of view is reversed: the numbers are not studied through the plane, but the plane through the numbers. A figure is described by the complex numbers of its points, and a geometric property by an equality between them.

The main tool is the complex exponential from the lesson on the complex exponential and Euler’s formula. For every real $\varphi$ , the number $e^{i\varphi}$ is the point of the circle of radius $1$ whose argument is $\varphi$ , and it has the following properties; the last one holds because $\cos(-\varphi) = \cos \varphi$ and $\sin(-\varphi) = -\sin \varphi$ :

e^{i\varphi} = \cos \varphi + i \sin \varphi, \qquad \abs{e^{i\varphi}} = 1, \qquad e^{i\alpha} e^{i\beta} = e^{i(\alpha + \beta)}, \qquad \conj{e^{i\varphi}} = e^{-i\varphi}

According to the lesson on the polar form of complex numbers, every nonzero complex number can be written $z = re^{i\theta}$ , with $r = \abs{z}$ and $\theta$ an argument of $z$ ; when two complex numbers are multiplied, their moduli are multiplied and their arguments are added, and when they are divided, their moduli are divided and their arguments are subtracted. In particular, multiplying $z$ by $e^{i\varphi}$ preserves its modulus and increases its argument by $\varphi$ : it rotates the point $z$ through the angle $\varphi$ about the origin, counterclockwise if $\varphi > 0$ . The product by $i = e^{\frac{i\pi}{2}}$ , studied in the Course I lesson on the complex numbers as a quarter-turn rotation, is the case $\varphi = \frac{\pi}{2}$ .

The points of a line segment can also be described with complex numbers. According to the Course I lesson on vectors and parametric equations of a line, the points of the line through $z_1$ and $z_2$ are those of the form $z_1 + t(z_2 - z_1)$ , with $t$ real, and those of the line segment joining them correspond to $0 \leq t \leq 1$ .

LemmaMidpoint and centroid in the complex plane

The midpoint of the line segment with endpoints $z_1$ and $z_2$ is half the sum $m = \frac{z_1 + z_2}{2}$ . The three medians of a triangle with vertices $z_1$ , $z_2$ and $z_3$ pass through the point $g = \frac{z_1 + z_2 + z_3}{3}$ , called the centroid, which lies on each median two thirds of the way from the vertex to the midpoint of the opposite side. The lemma was proved with vectors in the solved problem on the medians in the Course I lesson on vectors and parametric equations of a line; the proof below translates it into complex numbers. The figure follows it with the vertices $z_1 = -2 - i$ , $z_2 = 3 - i$ and $z_3 = 2i$ .

Proof

$\dato{m}{m = z_1 + \resaltar{\frac{1}{2}}(z_2 - z_1) = \resaltar{\frac{z_1 + z_2}{2}}}$
The point of the line segment that corresponds to $t = \frac{1}{2}$ is half the sum of the endpoints.
$m - z_1 = \resaltar{\frac{z_2 - z_1}{2}} = z_2 - m \rightarrow \resaltar{\abs{m - z_1} = \abs{z_2 - m}}$
It is at the same distance from both endpoints, because the two vectors joining it to them are equal: it is the midpoint.
$z_3 + \resaltar{\frac{2}{3}} \left( \frac{z_1 + z_2}{2} - z_3 \right) = \dato{g}{\resaltar{\frac{z_1 + z_2 + z_3}{3}}}$
On the median joining $z_3$ to the midpoint of the opposite side, the point that corresponds to $t = \frac{2}{3}$ is the following one.
$z_1 + \frac{2}{3} \left( \frac{z_2 + z_3}{2} - z_1 \right) = \resaltar{g} = z_2 + \frac{2}{3} \left( \frac{z_1 + z_3}{2} - z_2 \right)$
The expression obtained is symmetric in the three vertices; this being so, the same computation with either of the other two medians gives the same point, and all three pass through $g$ . With the vertices of the figure, $g = \frac{1}{3}$ .

Thus the midpoint is the average of two complex numbers, and the centroid, that of three.

In the rest of this lesson

Transformations of the plane
Translations and rotations
Dilations and similarity transformations
Reflections
The composition of two reflections
Collinearity, perpendicularity and angles
The equilateral triangle
Circles and the theorem on the circle of Apollonius
Solved problems
The fourth vertex of a square
Two loudspeakers and a circle of Apollonius
The composition of two rotations
Squares on two sides of a triangle
Napoleon’s theorem
Proposed problems
The reflection in an arbitrary line
Two reflections in parallel lines
The equilateral triangle without orientation
The center of Napoleon’s triangle
Van Aubel’s theorem