Euclidean Geometry - Homothety | Brilliant Math & Science Wiki

There are several ways to express a homothety. Switching between these different representations confer different benefits and can greatly simplify the problem, as will be observed later.

1) Geometric interpretation A homothety is just an expansion or contraction. The facts which apply to those situations also apply here. Here are examples of

• \( H (A, 2) \) applied to a triangle \( ABC: \)
• \( H \big( T, \frac{1}{2} \big) \) applied to a circle \( \Gamma \), where \(T\) is a point on the circumference:
• \( H(P, -1) \) applied to the line \(XY\), where \(P\) does not lie on \(XY:\)

2) Euclidean geometry (simplified)

• In the case that the center of homothety is the origin, an expansion by \(k\) would bring the point \( (x,y) \) to \( (kx, ky) \). This is the simplest case and often makes calculations very direct.
• In the case that the center of homothety is the point \( (a, b), \) an expansion by \(k\) would bring the point \( (x,y) \) to \( \big(k(x-a) + a, k(y-b) + b \big) \). This expression is slightly more complicated, which is why setting \( (a,b) = (0,0) \) is often more helpful.

3) Complex numbers In the argand plane, if the center of homothety is represented by the complex number \( p \), then an expasion by \(k\) would bring the complex number \(z\) to \( z' = p + k ( z - p ) \).

4) Vectors As stated above, if the center of homothety is represented by the vector \( \vec{P} \), then an expansion by \(k\) would bring the vector \( \vec{A} \) to \( \vec{A'} = \vec{P} + k \vec{ PA } \).

5) Abstract identification Given familiarity with the applications of homothety, we can identify a homothetic transformation and apply the results.