Cut the knot: learn to enjoy mathematics
A math books store at a unique math study site. Shopping at the store helps maintain the site. Thank you.
Learning Math Online
Sites for teachers
Sites for parents
Terms of use
Awards
Interactive Activities

CTK Exchange
CTK Wiki Math
CTK Insights - a blog
Math Help

III Millennium Olympiad

Games & Puzzles
What Is What
Arithmetic/Algebra
Geometry
Probability
Outline Mathematics
Make an Identity
Book Reviews
Stories for Young
Eye Opener
Analog Gadgets
Inventor's Paradox
Did you know?...
Proofs
Math as Language
Things Impossible
Visual Illusions
My Logo
Math Poll
Cut The Knot!
MSET99 Talk
Other Math sites
Front Page
Movie shortcuts
Personal info
Privacy Policy

Guest book
News sites

Recommend this site

Games to relax

Sites for teachers
Sites for parents

Education & Parenting

Manifesto  |  Bookstore  |  Contents  |  Amazon store  |  Term index  |  What changed?  |  Contact  |  Recommend
RSS Feed: Recent changes at CTK

Outline Mathematics
Geometry

Three Touching Circles

Consider the following problem:

 

Three circles S1, S2 and S3, touch pairwise in three distinct points. Show that the lines joining the point of tangency of S1 and S2 cross S3 again in points collinear with the center of S3, i.e., forming its diameter.



This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


Solution

Copyright © 1996-2009 Alexander Bogomolny

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Three circles S1, S2 and S3, touch pairwise in three distinct points. Show that the lines joining the point of tangency of S1 and S2 cross S3 again in points collinear with the center of S3, i.e., forming its .



This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


(In the text below, some words are omitted. These have been underlined. Click just above the line. See what happens.)

Let O, P, Q be the centers of circles S1, S2 and S3, respectively. Denote the tangency points of S2 and S3, S1 and S3, S1 and S2 as A, B, C. And, finally, let, D be the point of intersection of AC with S3 other than A, and E the point of intersection of BC with S3 other than B.

We know that CP|| and ||EQ. But O, C, P are . Therefore, D, Q, E are also collinear. It follows that DE is a diameter of S3.

Note that the statement holds both for external and internal tangency. In the latter case, the statement appears directly related to Lemma 1 from Archimedes' Book of Lemmas. Indeed, the significance of the role played by the tangency point C becomes transparent with an observation that it lies on common line of two diameters of two circles.

(The terms you met: Collinear points, Diameter of a circle)

References

  1. V. V. Prasolov, Problems in Planimetry, v 1, Nauka, Moscow, 1986, in Russian

Copyright © 1996-2009 Alexander Bogomolny

34222200Page copy protected against web site content infringement by Copyscape


Search:
Keywords:

Google
Web CTK