Inversion with respect to a circle

nntrkien · Hodge Conjecture Offline Joined: 24 Apr 2004 Posts: 60

Inversion is a strong tool to solve some kinds of problems. But I don't know very much about this method. Can everyone post some interesting problems about this ?
Thanks

**Posted:** Thu Aug 05, 2004 12:44 am

Johann Peter Dirichlet · Poincare Conjecture Offline Joined: 02 Apr 2004 Posts: 175

There is an article of Kin Li in Math.Excalibur:

http://www.math.ust.hk/excalibur/
http://www.math.ust.hk/excalibur/v9_n2.pdf

**Posted:** Thu Aug 05, 2004 2:46 am

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darij grinberg

In fact, inversion as a geometric transformation and as a method of solving problems is discussed very nicely by Alexander Bogomolny on the website http://www.cut-the-knot.org/Curriculum/Geometry/SymmetryInCircle.shtml and some other pages which are linked there.

See also http://www.artofproblemsolving.com/Forum/viewtopic.php?t=56658 and Kiran Kedlaya's Geometry Unbound.

Darij

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Check out the papers here:
www.math.utah.edu/mathcircle/notes/01-02.html

**Posted:** Wed Sep 14, 2005 8:01 am

Ashegh

i thinks before that u should learn about it and its theorems.i think if we post the theorems of inversion,it will help

u better. Wink

http://www.artofproblemsolving.com/Forum/viewtopic.php?t=51865

these links are some problems about inversion.

but i only could find my proofs for inversion. Embarassed

i will try and find the other inversion problems. Wink

**Posted:** Sat Oct 08, 2005 8:57 am

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neworder

Some problems to be solved by invesion:

1. Circles s1 and s2 are tangent externally in point K and tangent internally to the circle S in respectively A1 and A2. A line through K tangent to both circles s1 and s2 intersects S in point P. The line PA1 intersects s1 in point B1 and the line PA2 intersects s2 in point B2. Prove that line B1B2 is tangent to both s1 and s2. (it can be done without inversion, but inversion is very quick here)

2. Prove Ptolemy's inequality ( $AB*CD+BC*AD \geq AC*BD$ in a quadrilateral) using inversion.

3. Given points A,B and a circle O construct a circle passing through A and B and tangent to O.

4. Given a point A and cirlces O1, O2 construct a circle passing through A and tangent to both O1 and O2.

**Posted:** Thu Jan 19, 2006 7:06 am

puuhikki · Yang-Mills Theory Offline Joined: 06 Jun 2005 Posts: 849

What do you want to know about it? The definition of inversion is that if $C$ is an $n$ -dimensional ball $\mathbb{R}^{n}$ with radius $r$ and center $O$ and $P$ a point in $\mathbb{R}^{n}$ , then the inversion point of $P$ with respect to $C$ is a point $P'$ in ray $OP$ such that $OP\cdot OP'=r^{2}$ . Please ask more specific questions if you need more information.

**Posted:** Wed Oct 25, 2006 3:00 am

gemath · Yang-Mills Theory Offline Joined: 17 Mar 2006 Posts: 979

the most important fomula of invension is
If $A'=I_{k}(A),B'=I_{k}(B)$ then
$A'B'=\frac{kAB}{IA.IB}$

**Posted:** Wed Oct 25, 2006 3:29 am

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gabriel ponce

I would like to see some examples about inversion in olimpic problems.And the properties about quaht the inversion do with circles, lines, intersecting circles...

thanks

**Posted:** Wed Oct 25, 2006 9:09 am

sunchips

circles, tangents, etc. : very good representation using inversion

**Posted:** Thu Nov 30, 2006 1:38 pm

boxedexe

Logarithms is to Algebra as Inversion is to Geometry.

**Posted:** Thu Nov 30, 2006 3:23 pm

weiquan

**Posted:** Thu Nov 30, 2006 5:48 pm

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weiquan

Hmm.. coming to think of it, what about Mobius transformations? What are some useful properties of this mapping? I have not been able to find a nice book or website on this Sad

**Posted:** Wed Dec 06, 2006 2:09 am

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sunchips

**Posted:** Wed Dec 06, 2006 3:40 pm

weiquan

**Posted:** Wed Dec 06, 2006 7:47 pm

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weiquan

**Posted:** Mon Dec 11, 2006 6:31 pm

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tim1234133

There are loads by grobber, for example these:

http://www.artofproblemsolving.com/Forum/viewtopic.php?highlight=invert&t=106650
http://www.artofproblemsolving.com/Forum/viewtopic.php?highlight=invert&t=97487
http://www.artofproblemsolving.com/Forum/viewtopic.php?highlight=invert&t=41258
http://www.artofproblemsolving.com/Forum/viewtopic.php?highlight=invert&t=34049
http://www.artofproblemsolving.com/Forum/viewtopic.php?highlight=invert&t=32379
http://www.artofproblemsolving.com/Forum/viewtopic.php?highlight=invert&t=23756

**Posted:** Sat Jan 27, 2007 11:34 am

Jutaro

The best description of Möbius transformations I've ever read is in the book Visual Complex Analysis, of Tristan Needham. It's not that elementary, though. I don't know whether this transformations have applications in elementary geometry... maybe

**Posted:** Wed Feb 07, 2007 2:39 pm

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dondigo

http://en.wikipedia.org/wiki/Inversive_geometry
http://www.cut-the-knot.org/Curriculum/Geometry/SymmetryInCircle.shtml
http://www.math.ust.hk/excalibur/v9_n2.pdf

**Posted:** Wed Feb 07, 2007 11:32 pm

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stergiu

If it not already sent, I think this is also a nice pdf text on inversion !

http://www.imomath.com/tekstkut/inversion_ddj.pdf

Babis

**Posted:** Tue Dec 18, 2007 2:15 am