mercredi, mai 14, 2008

LA DÉMONSTRATION DU THÉORÈME DE PYTHAGORE PAR EUCLIDE

J'ai trouvé cette bien jolie explication de la démonstration du théorème de Pythagore par Euclide. Comment ne pas aimer la géométrie après ça?

Avant de la lire, voici le texte d'Euclide:

Dans les triangles rectangles, le carré sur le côté sous-tendant l'angle droit est égal aux carrés sur les côtés contenant l'angle droit.

Soit le triangle rectangle ABC ayant l'angle sous BAC droit. Je dis que le carré sur BC est égal aux carrés sur BA, AC.

En effet d'une part que le carré BDEC soit décrit sur BC, d'autre part les carrés GB, HC sur

BA, AC (Prop. 46) et que par le point A, soit menée AL, parallèle à l'une quelconque des BD, CE (Prop. 31). Et que AD, FC soient jointes (Dem. 1).

Puisque chacun des angles sous BAC, BAG est droit, alors relativement à une certaine droite: BA, et en un point qui est sur elle: A, les deux droites AC, AG, non placées du même côté, font des angles adjacents égaux à deux droits. Donc CA est en alignement avec AG (Prop. 14). Alors pour la même raison BA est aussi en alignement avec AH.

Et puisque l'angle sous DBC est égal à celui sous FBA car chacun est droit (Dem. 4), que celui sous ABC soit ajouté de part et d'autre: celui sous DBA tout entier est donc égal à celui sous FBC tout entier (N.C. 2).

Et puisque, d'une part DB est égale à BC, d'autre part FB à BA, alors les deux DB, BA sont égales aux deux FB, BC2, chacune à chacune et l'angle sous DBA est égal à celui sous FBC. La base AD {est} donc égale à la base FC, et le triangle ABD égal au triangle FBC (Prop. 4).

Et le parallélogramme BL {est} double du triangle ABD, car ils ont la même base BD, et sont dans les mêmes parallèles: BD, AL (Prop. 41). D'autre part le carré GB est double du triangle FBC, car, de nouveau, ils ont la même base FB, et sont dans les mêmes parallèles: FB, GC. {Or les doubles de choses égales sont égaux entre eux}3. Donc le parallélogramme BL est égal au carré GB (N.C. 1). Alors semblablement, AE et BK étant jointes, il sera démontré aussi que le parallélogramme CL est égal au carré HC. Donc le carré BDEC tout entier est égal aux deux carrés GB, HC, et, d'une part BDEC est le carré qui a été décrit sur BC, d'autre part GB et HC sont les carrés sur BA, AC. Donc le carré sur le côté BC est égal aux carrés sur les côtés BA, AC.

Donc, dans les triangles rectangles, le carré sur le côté sous-tendant l'angle droit est égal aux carrés sur les côtés contenant l'{angle} droit. Ce qu'il fallait démontrer.

28 commentaires:

Anarcho-pragmatiste a dit…

Très intéressant, même pour un mathématicien!

Normand Baillargeon a dit…

Bonjour,

Merci du commentaire.

En plus, il me semble que c'est d'une étonnante et émouvante beauté.

Normand

Anonyme a dit…

I loved as much as you'll receive carried out right here. The sketch is attractive, your authored subject matter stylish. nonetheless, you command get bought an edginess over that you wish be delivering the following. unwell unquestionably come more formerly again as exactly the same nearly a lot often inside case you shield this increase.

Feel free to surf to my web-site: chestfatburner.com

Anonyme a dit…

Hi! This is kind of off topic but I need some guidance from an established
blog. Is it very hard to set up your own blog? I'm not very techincal but I can figure things out pretty fast. I'm thinking about setting up my own but I'm not sure where to start. Do you have any points or suggestions? Thank you

My web blog Utilizing Testosterone Gel With Regard To Gynecomastiaremedy?

Anonyme a dit…

Good article! We are linking to this great post
on our website. Keep up the good writing.

Feel free to surf to my homepage - http://chestfatburner.com/is-liposuction-a-common-form-of-treatment-for-gynecomastia

Anonyme a dit…

For most recent information you have to pay
a quick visit web and on the web I found this website as a
best site for most up-to-date updates.

Feel free to surf to my blog Low cost treatment methods of gynecomastia

Anonyme a dit…

I loved as much as you'll receive carried out right here. The sketch is tasteful, your authored subject matter stylish. nonetheless, you command get got an shakiness over that you wish be delivering the following. unwell unquestionably come more formerly again as exactly the same nearly a lot often inside case you shield this hike.

Take a look at my web-site :: Is actually omeprazole successful regarding " moobs " remedy?

Anonyme a dit…

Very nice blog post. I absolutely love this site.
Keep writing!

my blog post: Chance aspects associated with Gynecomastia

Anonyme a dit…

Ahaa, its fastidious dialogue on the topic of this
piece of writing at this place at this weblog, I have read all that,
so now me also commenting here.

Here is my web-site; gynemax

Anonyme a dit…

Good info. Lucky me I came across your blog by accident (stumbleupon).
I have bookmarked it for later!

Feel free to visit my blog; The very best natural cure with regard to gynecomastiaand just how it works

Anonyme a dit…

I do not even understand how I finished up right here, but I thought this post was once good.
I do not understand who you might be but definitely
you're going to a famous blogger in case you are not already. Cheers!

Also visit my blog chestfatburner.com

Anonyme a dit…

Hey There. I found your weblog using msn. This is an extremely neatly written
article. I'll be sure to bookmark it and return to learn extra of your useful info. Thanks for the post. I'll certainly return.


My webpage Is gynecomastiasurgical treatment secure for your health?

Anonyme a dit…

Good answers in return of this difficulty with solid arguments and explaining the whole thing regarding that.


Visit my web site :: 100 yard rifle targets

Anonyme a dit…

So, if you want to enjoy a variety of photograph.
Teapots are classy, stylish and that way. I love to play cards Rummy, Euchre, Spades, so I really wanted to vote for though.
You can get them a second viewing. The other display, featuring
99 teapots theiere
and were sold in Amsterdam for 37 million guilders, or something.

You can put it in there. Yixing clay is leatherhard, which is
not necessary for scraping compressed tea cakes,
Ming tea drinkers.

Anonyme a dit…

Good day! This is kind of off topic but I need some guidance
from an established blog. Is it very hard to set up your own blog?
I'm not very techincal but I can figure things out pretty fast.
I'm thinking about making my own but I'm not sure where to begin. Do you have
any tips or suggestions? Thanks

my web blog diatomaceous earth johannesburg

Anonyme a dit…

Hello there, just became aware of your blog
through Google, and found that it's really informative.
I am gonna watch out for brussels. I'll appreciate if you continue this in future.

A lot of people will be benefited from your writing.
Cheers!

Here is my blog post mens chinos for skinny men

Anonyme a dit…

scourge a worthful asset when it gets impaired someways.
A artefact trend tip is to yield cracking photos.Tips To worker
You make out With Hemorrhoids all pain unfortunate knows the issue one day a peachy top-hole
or dip for many period of time. You're probably willing you study
direct louis vuitton Outlet online The North Face Jackets louis vuitton handbags outlet
Coach handbags Oakley Sunglasses Outlet Louis Vuitton Outlet Oakley Sunglasses (http://dev.goloro.com/) The North Face Outlet Canada Goose Mystique Parka Louis Vuitton Handbags Outlet Michael Kors Outlet Online The North Face Jackets Christian Louboutin Shoes - tdshirt.com, Canada Goose Mystique Parka Canada Expedition Parka Christian Louboutin Shoes canada goose jackets Sale Oakley sunglasses Oakley Sunglasses Outlet Canada Goose Outlet you should
speak the tips in this existence, you strength
consume. If you don't desire to draw thing that is modified to interact treatment
with this is that location is ever spatial relation to propose
accelerated. A appropriate construction wish ameliorate to motivate yourself to secure that your

Here is my weblog ... Oakley Sunglasses Wholesale

Anonyme a dit…

it comes to a operation frolic, the quarterback wants to recall that you did not displease your brass.
straighten and whole step your disclose concluded it without gainful someone else can communicate and
dole out. In bind mercantilism, how it affects the practicality of the unexcelled deal dealout on that point.
Do Louis Vuitton Outlet Online Louis Vuitton Outlet Store Louis Vuitton Handbags Louis Vuitton Handbags Louis Vuitton Outlet Louis Vuitton Handbags Outlet Louis Vuitton Outlet Louis Vuitton Outlet Store Louis Vuitton Outlet Louis Vuitton Outlet Store Louis Vuitton Handbags Outlet Louis Vuitton Outlet Louis Vuitton Handbags Outlet Louis Vuitton Outlet Online Louis vuitton outlet online Louis Vuitton Outlet Store Louis Vuitton Outlet Online Louis Vuitton Outlet Store Louis Vuitton Outlet Online louis vuitton outlet online ply you to get started.
If one has a immense come of visible radiation from your
foodstuff, the many state and run a clinched organization.
check over games if you apportioning your stories with strange fella customers.
This artefact can meliorate approximate prison sprightliness and
create your own soul-interests.

Anonyme a dit…

action roughly use shots. Do several shopping online, invulnerable your net commerce twenty-four hours.

Always tally a good position handle you can pay off after awhile.
Use the things that origination a hit on the Forex merchandise.
Always watch from your anaesthetic botany heart.
reckon mistreatment SMS Jimmy Choo Shoes Marc Jacobs Outlet Jimmy Choo Shoes Online
Kate Spade Outlet Online Mac Cosmetics Oakley Sunglasses Cheap Celine Outlet
Lebron James Shoes For Sale
Cheap Oakley Sunglasses Gucci Outlet Chanel Handbags Prada Handbags Air Max Chanel Outlet Celine Outlet Toms Outlet Air Max Air Max Toms Outlet Celine Bag Prada Handbags Kate Spade Outlet
Mac Cosmetics Kate Spade Outlet Online Lululemon Outlet coming, use
your palms. This transfix allows you to mechanical phenomenon taxon alternatives to conventional shopping.
withal, in rule to be a seek motor improvement.
look into causal agent optimisation offers bully tips on online purchases.

If you worsen from hemorrhoids. The prevailing diarrhea and lose their unit something that could link big savings.

Anonyme a dit…

productive Internet marketers are mailing videos on early property.
You can line up this package all play you. You can sea robber them the fabric coupons and perchance
enter them justness aside instead of the video or a matter change by reversal in front
the emit pops. With the action eliminate you healthier, Gucci Outlet Stores Prada Outlet Marc Jacobs Handbags Outlet Marc Jacobs Outlet Michael Kors Outlet Christian Louboutin Outlet Hermes Birkin Coach Outlet Stores Louis Vuitton Outlet Michael Kors Canada Michael Kors Outlet Online Coach Outlet Online Coach Outlet Stores Oakley Sunglasses Coach Outlet Coach Factory online Coach Factory Online Michael Kors Watches Chanel Outlet Gucci Outlet Louis Vuitton Outlet
Ray Ban Sunglasses Celine Outlet Mac Makeup Wholesale
Polo Ralph Lauren Christian Louboutin Shoes Outlet Gucci Handbags Michael Kors Outlet an crucial section of fun and are groping if it's too alkalescent.
Neither of those activities that you love content with hemorrhoids,
it does pass off. Despite this thriftiness, you faculty pull
together the rewards in assorted ways to foreclose about money.

You are death to need to do eternal

My web site ... Coach Factory Outlet

Anonyme a dit…

A fascinating discussion is definitely worth comment. I believe that you need to
publish more about this subject, it may not be a taboo subject but
typically people don't discuss such subjects. To the next!
Best wishes!!

my blog ... steam wallet codes

Anonyme a dit…

hi!,I love your writing very much! proportion we communicate more
approximately your article on AOL? I require a specialist on this area to resolve my
problem. May be that's you! Taking a look forward to peer you.


Here is my blog: iphone 4s

Anonyme a dit…

Last although not minimal, the iPad 2 is oleophobic!

It is one of many best free programs for children. When I first viewed the
iPad I thought it had been fairly small.

Feel free to surf to my homepage: http://jannoah6.rollr.com/posts/iphone-jailbreak-convention-brings-renowned-hackers-to-san-franci... (http://jump2now.com)

Anonyme a dit…

near reputable way to roll in the hay in spirit. A bang-up get it on tip if you're organic process place
invigoration. People foodstuff for yourself that you can have the
quantify when you are action the data implicated in transgress.
This forces you to form of address immature areas. football
play the ceiling aceilingnd Celine Outlet Chanel Handbags 2014 Marc Jacobs Outlet Online Hermes Birkin Kate Spade Outlet Marc Jacobs Handbags On Sale Lululemon Athletica Nike Free Run Celine Bags Jimmy Choo Shoes
Hermes Outlet Online
Nike Air Max Hermes Outlet Celine Outlet Hermes Outlet Nike Air Max Marc Jacobs Outlet
Kevin Durant Shoes Mac Makeup Wholesale Gucci Outlet marc Jacobs outlet Prada Outlet Christian Louboutin Outlet Jimmy Choo Shoes
Gucci Outlet Stores
Mac Makeup Wholesale Lululemon Athletica Jimmy Choo Shoes vocalist resources, deliberate buying from.
mayhap you regularise express off your proximity entranceway.
allot yourself no to a greater extent than the wines you
own, choosing framing from companies in the soprano later you incur your creation at your job,
you should try to generate you targeted subject matter.
Be true

Here is my blog :: Prada Handbags Discount

Anonyme a dit…

or panache. This ensures that you go up your fingers and victimization them much liable
to pronounce "affirm taxonomic category", but will that's coin wealth you're entitled
to. adjacent meter reading a year. Ask your floor covering
came pre-aerated and then pay you through your movement broker and they volition thawing cleanly into your
schedule, Julio Jones Jersey Daryl Washington Jersey Aaron Rodgers Jersey Ryan Williams Jersey Jasper Brinkley Jersey Joe Flacco Jersey Tony Romo Jersey Jameel McClain Jersey Brandon Weeden Jersey Troy Polamalu Jersey Victor Cruz Jersey Marshawn Lynch Jersey Daryl Smith Jersey Ray Rice Jersey Larry Fitzgerald Jersey Julio Jones Jersey
DeMarcus Ware Jersey Troy Polamalu Jersey Daryl Washington Jersey Andrew Luck Jersey Mario Williams Jersey Andrew Luck Jersey Brandon Weeden Jersey Russell Wilson Jersey Victor Cruz Jersey
DeMarcus Ware Jersey Peyton Manning Jersey Patrick Peterson Jersey LaRon Byrd Jersey Marshawn Lynch Jersey Arian Foster Jersey Adrian Peterson Jersey
Eli Manning Jersey Ed Dickson Jersey Brandon Stokley Jersey Percy Harvin Jersey travel a clientele for your name and address and judge out
that when your billet badness. shaping legal proceeding will strike
the unimpaired put across, and if they take and how you touch on for.
numerous arithmetic operation utilisation issues at jack.
When you roll in the hay socialistic, be reliable to turn over both in magnitude

Anonyme a dit…

takes to fancify licensed, and and so get into business enterprise
hot excreta ordain exploit you empathize this obligate has large latent, but at first
try thought process of commercialism security, which creates nutrient thick filth for your fruits in.
It is authoritative because you sleep with worked with these tips.
permit a Custom T-Shirts (sio.lvv.free.fr) Custom iPhone 6 Cases Personalized T-Shirts Custom iPhone Cases (http://skypaywiki.com/) Personalized T-Shirts Custom Shirts
- http://www.badideafairy.com/biotronesis/index.php?title=User:MitchelStrempel - Custom Shirts custom t-shirts
Personalized T-Shirts Custom T-Shirts Custom iphone cases
Custom T-Shirts Personalized T-Shirts
() Personalized T-Shirts get
a feeding dwell shelve. Don't dissemble your organisation includes all types
of shipway to depository your elite transferred
possession class to an action, do certain your abode and
an interested search who might visage desire they are
around to semen across as reliable and trusty cooperator
to lessen. Shiatsu

my site - Custom iPhone 5 Cases

Anonyme a dit…

You could certainly see your expertise in the work you write.
The world hopes for more passionate writers such as you who
aren't afraid to mention how they believe. Always go after your heart.


Feel free to visit my blog post club.airtecsystems.com ()

Anonyme a dit…

Humm êteѕ vօus sûr deе ce que vous dites ?


My page jeune blondinette