Complex Numbers

1. Algebraic Structure of Complex Numbers
2. Division of Complex Numbers
3. Useful Identities Among Complex Numbers
4. Useful Inequalities Among Complex Numbers
5. Trigonometric Form of Complex Numbers
6. Real and Complex Products of Complex Numbers
7. Complex Numbers and Geometry
8. Remarks on the History of Complex Numbers
   • First Geometric Interpretation of Negative and Complex Numbers
9. Complex Numbers: A Dynamic Tool
10. Cartesian Coordinate System
11. Fundamental Theorem of Algebra
12. Complex Number To a Complex Power May Be Real
13. One Can't Compare Two Complex Numbers

Complex Numbers and Geometry

Several features of complex numbers make them extremely useful in plane geometry. For example, the simplest way to express a spiral similarity in algebraic terms is by means of multiplication by a complex number. A spiral similarity with center at c, coefficient of dilation r and angle of rotation t is given by a simple formula

From (1) we can derive the equation of a line through two points, a and b say. Indeed, if the line contains both a and b, then it is parallel to the number b-a. Thus the equation becomes

or,

Given four complex numbers u, v, w, z. Then the following conditions are equivalent and each is satisfied iff the two segments zu and vw are perpendicular:

1. (u - v)/(w - z) is purely imaginary,
2. (u - v)/(w - z) + (u' - v')/(w' - z') = 0,
3. (u - v).(w - z) = 0,

where apostrophe denotes the conjugate of a complex number, and the dot stands for the real product of two numbers.

Given four complex numbers u, v, w, z. Then the following conditions are equivalent and each is satisfied iff the two segments zu and vw are parallel:

1. (u - v)/(w - z) is real,
2. (u - v)/(w - z) = (u' - v')/(w' - z'),
3. (u - v)×(w - z) = 0,

where the cross denotes the complex product of two numbers.

If v = z, we obtain the following condition for the collinearity of three points:

1. u, v, w are collinear,
2. (u - v)/(w - v) = (u' - v')/(w' - v'),
3. (u - v)×(w - v) = 0.

1. u, v, w, z are concyclic (or collinear),
2. (u - w)/(u - z) : (v - w)/(v - z) is real,
3. (u - w)/(u - z) : (v - w)/(v - z) = (u' - w')/(u' - z') : (v' - w')/(v' - z')
4. (uvwz) is real.

(uvwz) is a common shorthand of the double ratio in #2. The latter simply claims that the angles at u and v subtended by wz are either equal or their difference equals π modulo 2π.

In complex analysis, the cross-ratio (uvwz) is more often denoted (u,v;w,z) = (u - w)/(u - z) : (v - w)/(v - z). Collinearity is considered a special case of concyclicity.

As an exercise, you can verify a wonderful property of the cross-ratio. Let f(p), f(q), f(r), f(s) be four points on a line f(t) = (a + tb)/(1 + t). Then

Equilateral Triangles

For a positively oriented triangle A(a)B(b)C(c), the following conditions are equivalent

1. ABC is equilateral.
2. |a - b| = |b - c| = |c - a|.
3. a² + b² + c² = ab + bc + ca.
4. (b - a)/(c - b) = (c - b)/(a - b).
5. (z - a)^-1 + (z - b)^-1 + (z - c)^-1 = 0, where z = (a + b + c)/3.
6. (a + eb + e²c)(a + ec + e²b) = 0, where e = cos(2p/3) + i·sin(2p/3).

The following links point to a variety of applications of complex numbers in geometry:

Problems

1. 9-point Circle as a locus of concurrency
2. A Case of Similarity
3. A Property of Cubic Equations
4. All About Medians
5. Asymmetric Propeller
6. Bottema's Theorem
7. Cantor's Theorem
8. Center-circles and Their Chains
9. Clifford's Chain
10. Clifford's Lemma
11. Cycloids
12. Equilateral Triangles On Sides of a Parallelogram
13. Friendly Kiepert's Perspectors
14. Harmonic Ratio in Complex Domain
15. Hypocycloid Families
16. Iterations and the Mandelbrot Set
17. J. C. Maxwell's Theorem
18. Mandelbrot and Julia sets
19. Morley's Miracle: The Original Proof
20. Morley's Redux and More
21. Napoleon's and Douglas' Theorems
22. Napoleon's Propeller
23. Napoleon's Theorem
24. On Bottema's Shoulders II
25. Periodic Points of Quadratic Polynomials
26. Product of Diagonals in Regular N-gon
27. Remarkable Line in Cyclic Quadrilateral
28. Right Isosceles Triangles on Sides of a Quadrilateral
29. Three Isosceles Triangles
30. Thébault's Problem I
31. Thébault's Problem II
32. When a Triangle is Equilateral

References

1. T. Andreescu, D. Andrica, Complex Numbers From A to ... Z, Birkhäuser, 2006
2. C. W. Dodge, Euclidean Geometry and Transformations, Dover, 2004 (reprint of 1972 edition)
3. Liang-shin Hahn, Complex Numbers & Geometry, MAA, 1994
4. E. Landau, Foundations of Analisys, Chelsea Publ, 3^rd edition, 1966

Copyright © 1996-2008 Alexander Bogomolny