Geometry using complex numers

Xixas

Can anyone give me a link, a document or any other material, teaching of using complex numbers in geometry problems? Thank you in advance.

**Posted:** Sat Nov 20, 2004 5:29 am

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Kęstutis Česnavičius

billzhao

Try this:
http://www.math.ust.hk/excalibur/v9_n1.pdf

**Posted:** Sat Nov 20, 2004 6:13 am

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Yufei Zhao

Megus

I'm looking for books, articles, etc that would expand my knowledge of using complex numbers in geometry Mr. Green

I mean so texts beyond basics. I'll be glad to see any link

**Posted:** Sun Mar 13, 2005 7:39 am

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Przemyslaw Chojecki

kunny

How about this book?

Complex Numbers and Geometry written by Liang-shin Hahn

From the text of introducing author:

The purpose of this book is to demonstrate that complex numbers and geometry can be blended together beautifully,resulting in easy proofs and natural generalizations of many theorems in plane geometry---such as the theorems of Napoleon,Ptolemy---Euler,Simson,and Morley.

kunny

**Posted:** Sun Mar 13, 2005 7:50 am

Megus

kunny I have nothing against this book Smile

[I like the introduction] but there's no chance I get any title unless it is published on Internet freely - so that's why I'm asking for links.

Thanks for showing your interest though. Smile

**Posted:** Sun Mar 13, 2005 8:04 am

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Przemyslaw Chojecki

shobber

You can try 南盘江's post in the pre-olympiad section:

http://www.artofproblemsolving.com/Forum/viewtopic.php?t=41553

**Posted:** Fri Jun 24, 2005 3:26 am

bomb · Riemann Hypothesis Offline Joined: 29 Sep 2004 Posts: 363

What you are referring to I assume is the geometry of complex numbers.
Preparatory things include learning displacement vectors, inversion of geometry in complex plane, Mobius transformations etc.
I think a nice starting point is proving Ptolemy's inequality with it.

Bomb

**Posted:** Fri Jun 24, 2005 10:57 pm

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Man was born, born to be wild.
Man can climb, climb so high.
Man never want to die.....

stergiu

Well,

if you need such problems with solutions you can read the new(and very nice book) by Titu Andreescu and Dorin Andrica << Complex Numbers from A to Z >>, Edition birhauser.

You can find this book searching in

www.amazon.com

Put '' Titu andreescu '' , '' Dorin Andrica '' and you will have the book.
See also my post in High School section , Books and Journals. I have the link there. The best book solving difficult problems in geometry with comples numbers is << Modenov : Problems in geometry , edition MIR >> ,you can fie book only in big limbraries.

Of course there are also and other books but you can not buy them( they are not available now).

babis

**Posted:** Wed Jan 18, 2006 2:24 am

Tiks

90% of geometry problems can be solving with useing complex numbers Very Happy

(at laste for me Wink

),but for same problems that solutions is wery long Sad

.
For example,the 5th problem of the lest IMO I solved with conplex numbers,and I did have 6 point for that problem Very Happy

.(they take 1 point ,becaus in my solution was be samething juste a little not clear Blush

)

**Posted:** Thu Jan 19, 2006 9:32 am

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$\text{Tigran Sloyan}$

neworder

I know that - everything can be solved using complex numbers, but more often than not the "complex" solution is literally complex Wink

I meant not very hard problems that I could use to gain some skill in doing geometry using complex numbers. Any likns/problems?

**Posted:** Thu Jan 19, 2006 9:47 am

wojto111 · P versus NP Offline Joined: 18 Jan 2006 Posts: 42

We can write here some tricks Smile

. For example in complex numbers inversion of point $z$ in $(0,0)$ circle (with unit radius) is $\frac{1}{x\bar}$

**Posted:** Fri Jan 20, 2006 9:46 am

Tiks

**Posted:** Fri Jan 20, 2006 11:10 am

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$\text{Tigran Sloyan}$

Virgil Nicula

In the section "Geometry" are many solutions using the complex numbers for proposed problems. Read them!

**Posted:** Fri Jan 20, 2006 1:17 pm

stergiu

I have seen this book in the literatur , but can I find it in english or only in russian?

"Yaglom:Geometry and complex numbers"

babis

**Posted:** Sat Jan 21, 2006 2:48 pm

Tiks

**Posted:** Sun Jan 22, 2006 3:21 am

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$\text{Tigran Sloyan}$

Beat · Yang-Mills Theory Offline Joined: 12 Nov 2005 Posts: 662

Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$ . Define $A_s = A_{s-3}$ for all $s \geq 4$ . Construct a set of points $P_1, P_2, P_3, ...$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^{\circ}$ clockwise for $k = 0, 1, 2, ...$ . Prove that if $P_{1986} = P_0$ , then the triangle $A_1A_2A_3$ is equilateral. (IMO 86)

It is given a convex pentagon $ABCDE$ , such that $AB=BC=CA$ and $CD=DE=EC$ . Let $S$ be the medicentre of $\triangle ABC$ and $M$ and $N$ - the midpoints of $BD$ and $AE$ respectively. Prove that triangles $SME$ and $SND$ are similar.

A non-isosceles triangle $A_1A_2A_3$ has sides $a_1, a_2, a_3$ with ai opposite $A_i$ . $M_i$ is the midpoint of side ai and $T_i$ is the point where the incircle touches side $a_i$ . Denote by $S_i$ the reflection of $T_i$ in the interior bisector of $\angle A_i$ . Prove that the lines $M_1S_1$ , $M_2S_2$ and $M_3S_3$ are concurrent. (IMO 82)

Anyways, Tiks is so right.. Most of the geometry problems must have a solution with complex numbers

**Posted:** Sun Jan 22, 2006 4:27 am

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We are what we were

delta

if u know russian go to http://ilib.mirror0.mccme.ru/ from there you can download Yaglom's book(a classic)

if u dont know, take my advice: Try to learn!

**Posted:** Tue Mar 07, 2006 9:06 am

Xixas

There are two articles in the Mathematical Excalibur on this theme.

**Posted:** Fri Apr 07, 2006 11:27 am

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Kęstutis Česnavičius

stancioiu sorin

You can study the book written by Arthur Engel, from GIL, Romania.

**Posted:** Thu Sep 07, 2006 8:10 am

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$\ e^{\sqrt 3}-e\leq\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{e^{\cot^{2}x}}{\sin^{2}x}dx\leq\frac{e^{3}-e}{2}$

Rzeszut

**Posted:** Thu Sep 28, 2006 12:02 pm

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$\left(\int_{I}f(t)dt\right)^{k}= \int_{I^{k}}f(t_{1})\cdot\ldots\cdot f(t_{k}) dt_{1}\ldots dt_{k}$