Complex Numbers in Geometry — Examples & Problems

Collinearity of Three Points

Three collinear points A, B and C on a straight line

Let $A,B,C$ be points in the complex plane with complex coordinates $a,b,c$ , where $a\ne b$ . Then $A,B,C$ are collinear if and only if

\boxed{\frac{c-a}{b-a}\in\mathbb R.}

General Equation of a Line Through Two Complex Points

Line through two points A and B in the complex plane

For distinct complex numbers $a,b$ , the line through their corresponding points has equation

\boxed{z+\frac{a-b}{\overline b-\overline a}\,\overline z=\frac{a\overline b-\overline a b}{\overline b-\overline a}.}

Equivalently, $z+\lambda\overline z=\mu$ , where

\lambda=\frac{a-b}{\overline b-\overline a},\qquad \mu=\frac{a\overline b-\overline a b}{\overline b-\overline a}.

Here $|\lambda|=1$ and $\mu=\lambda\overline\mu$ .

Distance from a Point to a Line

Suppose the line $\ell$ has equation

\ell:\quad z+u\overline z=v,

where $|u|=1$ and $v=u\overline v$ . Then the distance from the point with complex coordinate $z_0$ to $\ell$ is

\boxed{d(z_0,\ell)=\frac{|z_0+u\overline{z_0}-v|}{2}.}

General Intersection Formula

If two nonparallel lines have equations

z+\lambda_1\overline z=\mu_1,\qquad z+\lambda_2\overline z=\mu_2,

then their intersection $P$ has complex coordinate

\boxed{p=\frac{\lambda_1\mu_2-\lambda_2\mu_1}{\lambda_1-\lambda_2}.}

The nonparallel condition is $\lambda_1\ne\lambda_2$ .

Intersection of Two Chord Lines of the Unit Circle

Unit circle with chords AB and CD intersecting at P

Let $A,B,C,D$ be four distinct points on the unit circle centered at the origin, with complex coordinates $a,b,c,d$ . Suppose the lines $AB$ and $CD$ intersect at $P$ . Then

\boxed{p=\frac{ab(c+d)-cd(a+b)}{ab-cd}.}

This requires $ab\ne cd$ , equivalently that the two lines are nonparallel.

Equality of Directed Angles

Let $A,B,C,D,E,F$ be points such that $A\ne B,C$ , $D\ne E,F$ . For directed angles

\alpha=\angle BAC,\qquad\beta=\angle EDF,

we have $\alpha\equiv\beta\pmod{180^\circ}$ if and only if

\boxed{\frac{(b-a)/(c-a)}{(e-d)/(f-d)}\in\mathbb R.}

Perpendicularity of Two Complex Vectors

Two perpendicular complex vectors with a right-angle mark

For nonzero complex numbers $z_1,z_2$ ,

\boxed{z_1\perp z_2\iff\frac{z_1}{z_2}\in i\mathbb R\iff\overline{\left(\frac{z_1}{z_2}\right)}=-\frac{z_1}{z_2}.}

Chord–Midpoint Perpendicularity

Circle centered at O with chord AB, its midpoint M, and OM perpendicular to AB

If $a,b$ are distinct points on a circle centered at the origin, then

\boxed{\frac{b-a}{b+a}\in i\mathbb R}

provided $a+b\ne0$ .
Geometrically, the chord $AB$ is perpendicular to the line joining the circle's center to the midpoint of $AB$ . If $a+b=0$ , the chord is a diameter and its midpoint is the center.

Circumcenter of a Triangle in the Complex Plane

Let $a,b,c$ be the complex coordinates of three noncollinear points $A,B,C$ . The complex coordinate $o$ of their circumcenter is

\boxed{o=\frac{(a-b)(|a|^2-|c|^2)-(a-c)(|a|^2-|b|^2)}{(a-b)(\overline a-\overline c)-(a-c)(\overline a-\overline b)}.}