Homothety

In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point. A homothety can be an enlargement (resulting figure is larger), identity transformation (resulting figure is congruent), or a contraction (resulting figure is smaller). A homothety with center $H$ and factor $k$ sends point $A$ to a point $A' \ni HA'=k\cdot HA$ This is denoted by $\mathcal{H}(H, k)$ .

The below picture is an example of a homothety that maps $ABCD$ into $A'B'C'D'$ . From the picture, $O$ is the center, and the factor is $\frac{OD'}{OD}$ . Additionally, $ABCD$ and $A'B'C'D'$ are homothetic with respect to $O$ . $[asy] pair A=(15,15),B=(30,15),C=(30,30),D=(15,30),a=(60,60),b=(120,60),c=(120,120),d=(60,120); dot((0,0)); draw(d--(0,0)--b,dotted); draw(c--(0,0),dotted); label("$O$",(0,0),SW); draw(A--B--C--D--A); dot(A); label("$A$",A,SW); dot(B); label("$B$",B,SE); dot(C); label("$C$",C,NE); dot(D); label("$D$",D,NW); draw(a--b--c--d--a); dot(a); label("$A'$",a,SW); dot(b); label("$B'$",b,SE); dot(c); label("$C'$",c,NE); dot(d); label("$D'$",d,NW); [/asy]$

Properties

A homothety with factor $-1$ is a $180^\circ$ rotation about the center.
A point, its image from the homothety, and the center of the homothety are all collinear.
- For a given polygon, the lines connecting each point to its corresponding point of a polygon that is homothetic with the given polygon are all concurrent.
The image of a line from a homothety is parallel to the original line.
The resulting image of a polygon from a homothety is similar to the original polygon.
- Likewise, the resulting image of a circle from a homothety is also a circle.

Usage

Homothety often means that we can use properties of similarity. Additionally, similarity (especially with circles) where parallel lines are used can indicate that homothety can be used, and homothety can be used to prove that three points are collinear.

Practice Problem

Four circles $\omega,$ $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$ with the same radius are drawn in the interior of triangle $ABC$ such that $\omega_{A}$ is tangent to sides $AB$ and $AC$ , $\omega_{B}$ to $BC$ and $BA$ , $\omega_{C}$ to $CA$ and $CB$ , and $\omega$ is externally tangent to $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$ . If the sides of triangle $ABC$ are $13,$ $14,$ and $15,$ the radius of $\omega$ can be represented in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n.$