🌀 Complex Plane Geometry

Use complex numbers to solve geometric problems. This advanced topic combines algebra and geometry in powerful ways.

🎯 Key Ideas

Complex Plane: Each complex number $z = a + bi$ corresponds to point $(a,b)$ in the plane.

Geometric Operations: Multiplication by complex numbers performs rotations and scaling.

Loci: Sets of complex numbers satisfying certain geometric conditions.

📐 Basic Geometric Interpretations

Distance

Distance between complex numbers $z_1$ and $z_2$: $$|z_1 - z_2| = \sqrt{(a_1-a_2)^2 + (b_1-b_2)^2}$$

Midpoint

Midpoint of $z_1$ and $z_2$: $$\frac{z_1 + z_2}{2}$$

Example: Find distance between $z_1 = 1 + 2i$ and $z_2 = 4 + 6i$

Solution:

  • $|z_1 - z_2| = |(1 + 2i) - (4 + 6i)| = |-3 - 4i| = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = 5$

🔄 Rotations and Scaling

Multiplication by $e^{i\theta}$

Multiplying by $e^{i\theta}$ rotates by angle $\theta$ counterclockwise.

Multiplication by $r$

Multiplying by real number $r$ scales by factor $r$.

Combined Transformation

$z \mapsto re^{i\theta}z$ rotates by $\theta$ and scales by $r$.

Example: Rotate $z = 1 + i$ by $90°$ counterclockwise

Solution:

  • $90° = \frac{\pi}{2}$ radians
  • $e^{i\pi/2} = \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} = 0 + i(1) = i$
  • $i(1 + i) = i + i^2 = i - 1 = -1 + i$

🎯 Loci in Complex Plane

Circle

$|z - z_0| = r$ represents circle with center $z_0$ and radius $r$.

Line

$\text{Re}(az) = c$ or $\text{Im}(az) = c$ represents lines.

Perpendicular Bisector

$|z - z_1| = |z - z_2|$ represents perpendicular bisector of segment from $z_1$ to $z_2$.

Example: Describe the locus $|z - 1| = 2$

Solution:

  • This represents all complex numbers $z$ whose distance from $1$ is $2$
  • Answer: Circle with center at $1$ and radius $2$

🔺 Triangle Properties

Equilateral Triangle

Points $z_1, z_2, z_3$ form equilateral triangle if and only if: $$z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$$

Centroid

Centroid of triangle with vertices $z_1, z_2, z_3$: $$\frac{z_1 + z_2 + z_3}{3}$$

Example: Show that $1, \omega, \omega^2$ form equilateral triangle where $\omega = e^{i2\pi/3}$

Solution:

  • Check the condition: $1^2 + \omega^2 + (\omega^2)^2 = 1 + \omega^2 + \omega^4$
  • Since $\omega^3 = 1$, we have $\omega^4 = \omega$
  • So: $1 + \omega^2 + \omega = 1 + \omega + \omega^2 = 0$ (sum of roots of unity)
  • Also: $1 \cdot \omega + \omega \cdot \omega^2 + \omega^2 \cdot 1 = \omega + \omega^3 + \omega^2 = \omega + 1 + \omega^2 = 0$
  • Since $0 = 0$, the points form an equilateral triangle.

🌟 Regular Polygons

Vertices of Regular $n$-gon

If one vertex is at $z_0$ and center at $z_c$, then vertices are: $$z_k = z_c + (z_0 - z_c)e^{i2\pi k/n}$$

for $k = 0, 1, 2, \ldots, n-1$.

Example: Find vertices of regular hexagon centered at origin with one vertex at $1$

Solution:

  • $z_c = 0$, $z_0 = 1$, $n = 6$
  • $z_k = 0 + (1 - 0)e^{i2\pi k/6} = e^{i\pi k/3}$
  • Vertices: $e^{i0} = 1$, $e^{i\pi/3}$, $e^{i2\pi/3}$, $e^{i\pi} = -1$, $e^{i4\pi/3}$, $e^{i5\pi/3}$

🔄 Similarity and Congruence

Similarity

Triangles with vertices $z_1, z_2, z_3$ and $w_1, w_2, w_3$ are similar if: $$\frac{w_2 - w_1}{w_3 - w_1} = \frac{z_2 - z_1}{z_3 - z_1}$$

Congruence

Triangles are congruent if the above ratio equals $1$.

🎯 AMC-Style Worked Example

Problem: In the complex plane, let $A = 1$, $B = i$, and $C$ be such that triangle $ABC$ is equilateral. Find all possible values of $C$.

Solution:

  1. Method 1 - Rotation: Rotate $B$ about $A$ by $60°$:

    • $C_1 = A + (B - A)e^{i\pi/3} = 1 + (i - 1)e^{i\pi/3}$
    • $= 1 + (i - 1)(\frac{1}{2} + i\frac{\sqrt{3}}{2})$
    • $= 1 + \frac{i - 1}{2} + i\frac{\sqrt{3}(i - 1)}{2}$
    • $= 1 + \frac{i - 1}{2} + \frac{\sqrt{3}(-1 - i)}{2}$
    • $= \frac{2 + i - 1 - \sqrt{3} - i\sqrt{3}}{2} = \frac{1 - \sqrt{3} + i(1 - \sqrt{3})}{2}$

  2. Method 2 - Rotation by $-60°$: Rotate $B$ about $A$ by $-60°$:

    • $C_2 = A + (B - A)e^{-i\pi/3} = 1 + (i - 1)e^{-i\pi/3}$
    • Similar calculation gives $C_2 = \frac{1 + \sqrt{3} + i(1 + \sqrt{3})}{2}$

Answer: $C = \frac{1 \pm \sqrt{3} + i(1 \pm \sqrt{3})}{2}$

🔍 Common Traps & Fixes

Trap: Wrong rotation direction

Fix: $e^{i\theta}$ rotates counterclockwise; $e^{-i\theta}$ rotates clockwise.

Trap: Forgetting both rotation directions

Fix: For equilateral triangles, there are usually two possible positions.

Trap: Confusing distance and argument

Fix: $|z_1 - z_2|$ is distance; $\arg(z_1 - z_2)$ is direction.

Trap: Wrong locus interpretation

Fix: $|z - z_0| = r$ is circle; $|z - z_1| = |z - z_2|$ is line.

📋 Quick Reference

Geometric Operations

  • Distance: $|z_1 - z_2|$
  • Midpoint: $\frac{z_1 + z_2}{2}$
  • Rotation by $\theta$: multiply by $e^{i\theta}$
  • Scaling by $r$: multiply by $r$

Common Loci

  • Circle: $|z - z_0| = r$
  • Line: $|z - z_1| = |z - z_2|$ (perpendicular bisector)
  • Ray: $\arg(z - z_0) = \theta$

Triangle Properties

  • Centroid: $\frac{z_1 + z_2 + z_3}{3}$
  • Equilateral condition: $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$

Prev: Complex Numbers
Next: Sequences & Series
Back to: Topic Guides