# 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

f(z) = r(z - c)(cos(t) + i·sin(t)) + c.

### Circle

A particularly simple equation is that of a circle:

{z: |z - a| = r},

is the circle with radius r and center a. By squaring that equation we obtain

(z - a)(z' - a') = r²

or

zz' - (za' + z'a) + (aa' - r²) = 0.

and finally

zz' - (za' + z'a) + s = 0,

where s is a real number. The circle is centered at a and has the radius r = √aa' - s, provided the root is real.

This representation of the circle is more convenient in some respects. For example, we may immediately check that the transformation

1/w × 1/w' - (a'/w + a/w') + s = 0

which, if multiplied by ww', leads to

ww' - (wb' + w'b) + t = 0,

where b = a'/s and t = 1/s, an equation in the same form.

Letting a = α + iβ yields yet another form of essentially same equation:

zz' - α(z + z') - iβ(z - z') + s = 0,

where α and β are both real. Yet the most general form of the equation is this

Azz' + Bz + Cz' + D = 0,

which represents a circle if A and D are both real, whilst B and C are complex and conjugate. For

### Straight Line

A straight line through point (complex number) a and parallel to the vector (another complex number) v is defined by

(1)

f(t) = a + tv,

where t a real number. The line is the set {f(t): -∞ < t ≤ ∞} to show that any line contains a point at infinity. (The values at ±∞ are the same, so we chose just one of them, virtually arbitrarily.)

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

f(t) = a + t(b - a),

or,

f(t) | = (1 - t)a + tb |

= (1 - t)a + tb | |

= sa + tb, where s = 1 - t, | |

= (sa + tb) / (s + t), since s + t = 1, | |

= (a + rb) / (1 + r), |

where r = t/s = t / (1 - t). The latter defines a hyperbola in the

(2)

f(r) = (a + rb) / (1 + r).

The point at infinity is now obtained for

### Orthogonality

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

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

### Collinearity

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

- (u - v)/(w - z) is real,
- (u - v)/(w - z) = (u' - v')/(w' - z'),
- (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:

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

### Concyclicity

Given four complex numbers u, v, w, z. Then the following conditions are equivalent:

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

(uvwz) is a common shorthand of the double (cross-) 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

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

(f(p), f(q); f(r), f(s)) = (p, q; r, s).

### Similarity

Given two triangles A(a)B(b)C(c) and A_{1}(a_{1})B_{1}(b_{1})C_{1}(c_{1}). Then the following are equivalent"

- The triangles are similar and have the same orientation,
- (b
_{1}- a_{1})/(c_{1}- a_{1}) = (b - a)/(c - a).

Also,

- The triangles are similar and have different orientations,
- (b
_{1}- a_{1})/(c_{1}- a_{1}) = (b' - a')/(c' - a').

### Equilateral Triangles

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

- ABC is equilateral.
- |a - b| = |b - c| = |c - a|.
- a² + b² + c² = ab + bc + ca.
- (b - a)/(c - b) = (c - b)/(a - b).
- (z - a)
^{-1}+ (z - b)^{-1}+ (z - c)^{-1}= 0, wherez = (a + b + c)/3. - (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

- 9-point Circle as a locus of concurrency
- A Case of Similarity
- A Property of Cubic Equations
- All About Medians
- An Unexpected Pair of Similar Triangles Which Are Equal
- Asymmetric Propeller
- Bisector of an imaginary angle may be real
- Bottema's Theorem
- Cantor's Theorem
- Center-circles and Their Chains
- Clifford's Chain
- Clifford's Lemma
- Cycloids
- Equilateral Triangle on Angle Bisectors
- Equilateral Triangles On Sides of a Parallelogram
- Fermat's Hexagon
- Five Squares in Complex Numbers
- Four Hinged Squares
- Friendly Kiepert's Perspectors
- Harmonic Ratio in Complex Domain
- Hypocycloid Families
- Iterations and the Mandelbrot Set
- J. C. Maxwell's Theorem
- Kiepert's Centroid
- Kiepert's Triangles Graduate to Ears of Arbitrary Shape
- Mandelbrot and Julia sets
- Morley's Miracle: The Original Proof
- Morley's Redux and More
- Napoleon's and Douglas' Theorems
- Napoleon's Propeller
- Napoleon's Relatives
- Napoleon's Theorem
- On Bottema's Shoulders II
- Periodic Points of Quadratic Polynomials
- Product of Diagonals in Regular N-gon
- Remarkable Line in Cyclic Quadrilateral
- Right Isosceles Triangles on Sides of a Quadrilateral
- Spiral Similarity Leads to Equilateral Triangle
- Three Isosceles Triangles
- Thébault's Problem I
- Thébault's Problem II
- There is no Difference Between Equilateral Triangles
- Two Pencils of Parallel Lines
- When a Triangle is Equilateral

### References

- T. Andreescu, D. Andrica,
*Complex Numbers From A to ... Z*, Birkhäuser, 2006 - C. W. Dodge,
*Euclidean Geometry and Transformations*, Dover, 2004 (reprint of 1972 edition) - Liang-shin Hahn,
*Complex Numbers & Geometry*, MAA, 1994 - E. Landau,
*Foundations of Analisys*, Chelsea Publ, 3^{rd}edition, 1966 - D. Pedoe,
*Geometry: A Comprehensive Course*, Dover, 1988

### Complex Numbers

- Algebraic Structure of Complex Numbers
- Division of Complex Numbers
- Useful Identities Among Complex Numbers
- Useful Inequalities Among Complex Numbers
- Trigonometric Form of Complex Numbers
- Real and Complex Products of Complex Numbers
- Complex Numbers and Geometry
- Plane Isometries As Complex Functions
- Remarks on the History of Complex Numbers
- Complex Numbers: an Interactive Gizmo
- Cartesian Coordinate System
- Fundamental Theorem of Algebra
- Complex Number To a Complex Power May Be Real
- One can't compare two complex numbers
- Riemann Sphere and Möbius Transformation
- Problems

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