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Difference between revisions of "Homothety"
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In [[mathematics]], a '''homothety''' (or '''homothecy''') is a 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>.
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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 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>.
 
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>.
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*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>
 
*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>
  
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([[2007 AIME II/Problem 15|Source]])
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([[2007 AIME II Problems/Problem 15|Source]])
  
 
==References==
 
==References==

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

(Source)

References

See Also

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