Difference between revisions of "Homothety"

Latest revision as of 12:51, 2 September 2024

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 $m+n.$

(Source)

References

https://brilliant.org/wiki/euclidean-geometry-homothety/ (contains sample problems and related proofs)
https://euclid.ucc.ie/mathenr/IMOTraining/2018JuneCamp/HomothetyExcalibur.pdf

@@ Line 1: / Line 1: @@
-In [[mathematics]], a '''homothety''' (or '''homothecy''') is a transformation of space which dilates distances with respect to a fixed point. Such a transformation is also called an '''enlargement'''. A homothety with center <math>H</math> and factor <math>k</math> sends point <math>A</math> to a point <math>A' \ni HA'=k\cdot HA</math> This is denoted by <math>\mathcal{H}(H, k)</math>.
+In [[mathematics]], a '''homothety''' (or '''homothecy''') is a [[transformations|transformation]] of space which [[dilation|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 <math>H</math> and factor <math>k</math> sends point <math>A</math> to a point <math>A' \ni HA'=k\cdot HA</math> This is denoted by <math>\mathcal{H}(H, k)</math>.
-The following observations are noteworthy:
-Under a homothety, parallel lines map onto parallel lines.
-Homothety preserves angles.
-Homothety preserves orientation.
-Every homothety has an inverse, viz., HO, k-1 = HO, 1/k. In other words
-HO, k(HO, 1/k(P)) = P = HO, 1/k(HO, k(P)).
-Every homothety with k different from 1 has one and only one fixed point - the center O. Every line through O is also fixed although not pointwise. For k = 1, homothety is the identity transformation.
+The below picture is an example of a homothety that maps <math>ABCD</math> into <math>A'B'C'D'</math>.  From the picture, <math>O</math> is the center, and the factor is <math>\frac{OD'}{OD}</math>.  Additionally, <math>ABCD</math> and <math>A'B'C'D'</math> are homothetic with respect to <math>O</math>.
-Successive applications of homotheties with coefficients k1 and k2 is either a homothety with coefficient k1k2, if the latter differs from 1, or a parallel translation, otherwise. In the former case, the centers of the three homotheties are collinear. In the latter case, the translation is parallel to the line joining the centers of the two homotheties.
+<asy>
-The product of homotheties is not in general commutative. Two homotheties with a common center commute as a matter of course.
+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);
-== See Also ==
+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 <math>-1</math> is a <math>180^\circ</math> 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 [[circle]]s <math>\omega,</math> <math>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math> with the same [[radius]] are drawn in the interior of [[triangle]] <math>ABC</math> such that <math>\omega_{A}</math> is [[tangent]] to sides <math>AB</math> and <math>AC</math>, <math>\omega_{B}</math> to <math>BC</math> and <math>BA</math>, <math>\omega_{C}</math> to <math>CA</math> and <math>CB</math>, and <math>\omega</math> is [[externally tangent]] to <math>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math>. If the sides of triangle <math>ABC</math> are <math>13,</math> <math>14,</math> and <math>15,</math> the radius of <math>\omega</math> can be represented in the form <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are [[relatively prime]] positive integers. Find <math>m+n.</math>
+([[2007 AIME II Problems/Problem 15|Source]])
+==References==
+* https://brilliant.org/wiki/euclidean-geometry-homothety/ (contains sample problems and related proofs)
+* https://euclid.ucc.ie/mathenr/IMOTraining/2018JuneCamp/HomothetyExcalibur.pdf
+==See Also==
 * [[Dilation]]
+* [[Similarity]]
-{{stub}}
 [[Category:Definition]]
 [[Category:Geometry]]

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