Editing Complex number (section)

===Geometry===
====Shapes====
Three [[collinearity|non-collinear]] points <math>u, v, w</math> in the plane determine the [[Shape#Similarity classes|shape]] of the triangle <math>\{u, v, w\}</math>. Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as
<math display=block>S(u, v, w) = \frac {u - w}{u - v}. </math>
The shape <math>S</math> of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an [[affine transformation]]), corresponding to the intuitive notion of shape, and describing [[similarity (geometry)|similarity]]. Thus each triangle <math>\{u, v, w\}</math> is in a [[shape#Similarity classes|similarity class]] of triangles with the same shape.<ref>{{cite journal |last=Lester |first=J.A. |title=Triangles I: Shapes |journal=[[Aequationes Mathematicae]] |volume=52 |pages=30–54 |year=1994 |doi=10.1007/BF01818325 |s2cid=121095307}}</ref>

====Fractal geometry====
[[File:Mandelset hires.png|right|thumb|The Mandelbrot set with the real and imaginary axes labeled.]]
The [[Mandelbrot set]] is a popular example of a fractal formed on the complex plane. It is defined by plotting every location <math>c</math> where iterating the sequence <math>f_c(z)=z^2+c</math> does not [[diverge (stability theory)|diverge]] when [[Iteration|iterated]] infinitely. Similarly, [[Julia set]]s have the same rules, except where <math>c</math> remains constant.

====Triangles====
Every triangle has a unique [[Steiner inellipse]] – an [[ellipse]] inside the triangle and tangent to the midpoints of the three sides of the triangle. The [[Focus (geometry)|foci]] of a triangle's Steiner inellipse can be found as follows, according to [[Marden's theorem]]:<ref>{{cite journal |last1=Kalman|first1=Dan|title=An Elementary Proof of Marden's Theorem |url=http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1 |journal=[[American Mathematical Monthly]] |volume=115 |issue=4 |pages=330–38 |year=2008a |doi=10.1080/00029890.2008.11920532 |s2cid=13222698 |issn=0002-9890 |access-date=1 January 2012 |archive-url=https://web.archive.org/web/20120308104622/http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1 |archive-date=8 March 2012|url-status=live}}</ref><ref>{{cite journal |last1=Kalman |first1=Dan |title=The Most Marvelous Theorem in Mathematics |url=http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663 |journal=[[Journal of Online Mathematics and Its Applications]] |year=2008b |access-date=1 January 2012|archive-url=https://web.archive.org/web/20120208014954/http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663 |archive-date=8 February 2012 |url-status=live}}</ref> Denote the triangle's vertices in the complex plane as {{math|1=''a'' = ''x''<sub>''A''</sub> + ''y''<sub>''A''</sub>''i''}}, {{math|1=''b'' = ''x''<sub>''B''</sub> + ''y''<sub>''B''</sub>''i''}}, and {{math|1=''c'' = ''x''<sub>''C''</sub> + ''y''<sub>''C''</sub>''i''}}. Write the [[cubic equation]] <math>(x-a)(x-b)(x-c)=0</math>, take its derivative, and equate the (quadratic) derivative to zero. Marden's theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.