Editing Homothety (section)

=== Mapping lines, line segments and angles ===
A homothety has the following properties:
* A ''line'' is mapped onto a parallel line. Hence: ''angles'' remain unchanged.
* The ''ratio of two line segments'' is preserved.
Both properties show: 
* A homothety is a ''[[Similarity (geometry)|similarity]]''.

''Derivation of the properties:''
In order to make calculations easy it is assumed that the center <math>S</math>  is the origin: <math>\mathbf x \to k\mathbf x</math>. A line  <math>g</math> with parametric representation <math>\mathbf x=\mathbf p +t\mathbf v</math> is mapped onto the point set <math>g'</math> with equation <math>\mathbf x=k(\mathbf p+t\mathbf v)= k\mathbf p+tk\mathbf v</math>, which is a line parallel to <math>g</math>.

The distance of two points <math>P:\mathbf p,\;Q:\mathbf q</math> is <math>|\mathbf p -\mathbf q|</math> and <math>|k\mathbf p -k\mathbf q|=|k||\mathbf p-\mathbf q|</math> the distance between their images. Hence, the ''ratio'' (quotient) of two line segments remains unchanged.

In case of <math>S\ne O</math> the calculation is analogous but a little extensive.

Consequences: A triangle is mapped on a [[Similarity (geometry)|similar]] one. The homothetic image of a [[circle]] is a circle. The image of an [[ellipse]] is a similar one. i.e. the ratio of the two axes is unchanged.

[[File:Zentr-streck-T-S-e.svg|thumb|upright=0.8|With [[intercept theorem]]]]