# 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 w = f(z) = 1/z maps circles onto circles. Indeed, substituting z = 1/w we get

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 A = 0, the equation represents a straight line.

### 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 (t, r) plane so that r takes exactly the same values as t. In terms if thus defined r the straight line through a and b has the equation

(2)

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

The point at infinity is now obtained for r = -1. a = f(0), b = f(∞), (a + b)/2 = f(1).

### 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:

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.

### 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:

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.

### Concyclicity

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

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 (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 (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

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

### Similarity

Given two triangles A(a)B(b)C(c) and A1(a1)B1(b1)C1(c1). Then the following are equivalent"

1. The triangles are similar and have the same orientation,
2. (b1 - a1)/(c1 - a1) = (b - a)/(c - a).

Also,

1. The triangles are similar and have different orientations,
2. (b1 - a1)/(c1 - a1) = (b' - a')/(c' - a').

### 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:

### 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, 3rd edition, 1966
5. D. Pedoe, Geometry: A Comprehensive Course, Dover, 1988 ### Complex Numbers 