In plane geometry, complex numbers can be used to represent points, and thus other geometric objects as well such as lines, circles, and polygons. They are somewhat similar to Cartesian coordinates in the sense that they are used to algebraically prove geometric results, but they are especially useful in proving results involving circles and/or regular polygons (unlike Cartesian coordinates, which are useful for proving results involving lines).
A point in the plane can be represented by a complex number
which corresponds to the Cartesian point . Therefore, the -axis is renamed the real axis and the -axis is renamed the imaginary axis, or imaginary line. This can also be converted into a polar coordinate , which represents the complex number
which, intuitively speaking, means rotating the point an angle of about the origin. By Euler's formula, this is equivalent to
which means that the polar coordinate corresponds to the Cartesian coordinate
Incidentally, this immediately illustrates why complex numbers are so useful for circles and regular polygons: these involve heavy use of rotations, which are easily expressed using complex numbers.
In particular, a rotation of about the origin sends for all
In comparison, rotating Cartesian coordinates involves heavy calculation and (generally) an ugly result.
Additionally, each point has an associated conjugate . It satisfies the properties
Geometrically, the conjugate can be thought of as the reflection over the real axis. This implies two useful facts: if is real, , and if is pure imaginary, .
In this and the following sections, a capital letter denotes a point and the analogous lowercase letter denotes the complex number associated with it.
Though lines are less nice in complex geometry than they are in coordinate geometry, they still have a nice characterization:
The points are collinear if and only if is real, or equivalently, if and only if
There are two similar results involving lines. This is the one for parallel lines:
Lines and are parallel if and only if is real, or equivalently, if and only if
The following is the result for perpendicular lines:
Lines and are perpendicular if and only if is pure imaginary, or equivalently, if and only if
Additionally, there is a nice expression of reflection and projection in complex numbers:
Let be the reflection of over . Then
and the projection of onto is .
However, it is easy to express the intersection of two lines in Cartesian coordinates. In complex coordinates, this is not quite the case:
Lines and intersect at the point
which is impractical to use in all but a few specific situations (e.g. when one of the points is at 0).
The unit circle is of special interest in the complex plane, as points on the complex plane satisfy the key property that
which is a consequence of the fact that . This means that
in general, complex geometry is most useful when there is a primary circle in the problem that can be set to the unit circle.
For instance, some of the formulas from the previous section become significantly simpler. Reflection and projection, for instance, simplify nicely:
If lie on the unit circle, the reflection of across is . The projection of onto is thus .
Also, the intersection formula becomes practical to use:
If lie on the unit circle, lines and intersect at
Triangles in complex geometry are extremely nice when they can be placed on the unit circle; this is generally possible, by setting the triangle's circumcircle to the unit circle. This immediately implies the following obvious result:
Suppose lie on the unit circle. Then the circumcenter of is 0.
This is because the circumcenter of coincides with the center of the unit circle.
The centroid is also easy to compute:
Suppose lie on the unit circle. Then the centroid of is .
This also illustrates the similarities between complex numbers and vectors.
More interestingly, we have the following theorem:
Suppose lie on the unit circle. Then the orthocenter of is
Let . Recall from the "lines" section that is perpendicular to if and only if is pure imaginary. This is equal to since . Then
Since are on the unit circle, and . So
which implies . From the intro section, this implies that is pure imaginary, so is perpendicular to .
By similar logic, is perpendicular to and to , so is the orthocenter, as desired.
It is also possible to find the incenter, though it is considerably more involved:
Suppose lie on the unit circle, and let be the incenter. Let be the feet of the angle bisectors from respectively. Then there exist complex numbers such that . Then
There are two other properties worth noting before attempting some problems. The first is the tangent line through the unit circle:
Let lie on the unit circle. Then lies on the tangent through if and only if
This is especially useful in the case of two tangents:
Let be points on the unit circle. Their tangents meet at the point the harmonic mean of and .
Let be the intersection point. Then and , so
Thus, . Since lie on the unit circle, and , so as desired.
The second result is a condition on cyclic quadrilaterals:
Points lie on a circle if and only if
This section contains Olympiad problems as examples, using the results of the previous sections.
A point is taken inside a circle. For every chord of the circle passing through consider the intersection point of the two tangents at the endpoints of the chord. Find the locus of these intersection points.
WLOG assume that is on the real axis. Let be the endpoints of a chord passing through . From the previous section, the tangents through and intersect at . It is also true since are collinear, that
Now note that
so must lie on the vertical line through .
For any point on this line, connecting the two tangents from the point to the unit circle at and allows the above steps to be reversed, so every point on this line works; hence, the desired locus is this line. More formally, the locus is a line perpendicular to that is a distance from .
The following application of what we have learnt illustrates the fact that complex numbers are more than a tool to solve or "bash" geometry problems that have otherwise "beautiful" synthetic solutions, they often lead to the most beautiful and unexpected of solutions.
Consider a polygonal line such that , all measured clockwise. If , and cannot coincide.
Let us consider complex coordinates with origin at and let the line be the x-axis. Let be the angle between any two consecutive segments and let be the lengths of the segments. If we set , then the coordinate of is . We must prove that this number is not equal to zero.
Using the Abel Summation lemma, we obtain,
If is zero, then this quantity is a strictly positive real number, and we are done.
If not, multiply by to get . This expression cannot be zero. Indeed, since , by the triangle inequality, we have,
The conclusion follows.