The Devil's Triangle
Contents
[hide]Definition
Generalized Wooga Looga Theorem (The Devil's Triangle)
For any triangle , let and be points on and respectively. The Generalized Wooga Looga Theorem (Gwoologth) or the Devil's Triangle Theorem states that if and , then .
(*Simplification found by @Gogobao)
Proofs
Proof 1
Proof by CoolJupiter:
We have the following ratios: .
Now notice that .
We attempt to find the area of each of the smaller triangles.
Notice that using the ratios derived earlier.
Similarly, and .
Thus, .
Finally, we have .
~@CoolJupiter
Proof 2
Proof by math_comb01 Apply Barycentrics . Then . also
In the barycentrics, the area formula is where is a random triangle and is the reference triangle. Using this, we ===
~@Math_comb01
Other Remarks
This theorem is a generalization of the Wooga Looga Theorem, which @RedFireTruck claims to have "rediscovered". The link to the theorem can be found here: https://artofproblemsolving.com/wiki/index.php/Wooga_Looga_Theorem
Essentially, Wooga Looga is a special case of this, specifically when .
Testimonials
This is Routh's theorem isn't it~ Ilovepizza2020
Wow this generalization of my theorem is amazing. good job. - Foogle and Hoogle, Members of the Ooga Booga Tribe of The Caveman Society
trivial by but ok ~ bissue
"Very nice theorem" - RedFireTruck (talk) 12:12, 1 February 2021 (EST)
[hide=testimonials for the wooga looga theorem which is a special case of the devils triangle]Pogpr0 = wooga looga - Ladka13 The Wooga Looga Theorem can be used to prove many problems and should be a part of any geometry textbook. ~ilp2020
Thanks for rediscovering our theorem RedFireTruck - Foogle and Hoogle of The Ooga Booga Tribe of The Caveman Society
Franzlist is wooga looga howsopro - volkie boy
this is in fact a pretty sensible theorem. Nothing to be so excited about, though. ~DofL
The Wooga Looga Theorem is EPIC POGGERS WHOLESOME 100 KEANU CHUNGUS AMAZING SKILL THEOREM!!!!!1!!!111111 -centslordm
The Wooga Looga Theorem is amazing and can be applied to so many problems and should be taught in every school. - RedFireTruck (talk) 11:00, 1 February 2021 (EST)
The Wooga Looga Theorem is the best. -aaja3427
The Wooga Looga Theorem is needed for everything and it is great-hi..
The Wooga Looga Theorem was made by the author of the 5th Testimonial, RedFireTruck, which means they are the ooga booga tribe... proof: go to https://www.youtube.com/channel/UC50E9TuLIMWbOPUX45xZPaQ and click "about". now copy and paste the aops URL. you got RedFireTruck! Great Job! now go check out his thread for post milestones, https://artofproblemsolving.com/community/c3h2319596, and give him a friend request! -FPT
This theorem has helped me with school and I am no longer failing my math class. -mchang
I, who [u]rarely[\u] edits the AoPS Wiki, has edited this to show how amazing this theorem is! The Wooga Looga theorem has actually helped me on a school test, a math competition, and more! My teacher got upset at me for not doing it the way I she taught it though - ChrisalonaLiverspur
"I can't believe AoPS books don't have this amazing theorem. If you need help with math, you can depend on caveman." ~CoolJupiter
Before the Wooga Looga Theorem, I had NO IDEA how to solve any hard geo. But, now that I've learned it, I can solve hard geo in 7 seconds ~ ilp2020 (2nd testimonial by me)
Too powerful... ~franzliszt
The Wooga Looga Theorem is so pro ~ ac142931
It is so epic and awesome that it will blow the minds of people if they saw this ~ ac142931(2nd testimonial by me)
This theorem changed my life... ~ samrocksnature
Math competitions need to ban the use of the Wooga Looga Theorem, it's just too good. ~ jasperE3
It actually can be. I never thought I'd say this, but the Wooga Looga theorem is a legit theorem. ~ jasperE3
This is franzliszt and I endorse this theorem. ~franzliszt
This theorem is too OP. ~bestzack66
This is amazing! However much it looks like a joke, it is a legitimate - and powerful - theorem. -Supernova283
Wooga Looga Theorem is extremely useful. Someone needs to make a handout on this so everyone can obtain the power of Wooga Looga ~RP3.1415
The Wooga Looga cavemen were way ahead of their time. Good job (dead) guys! -HIA2020
It's like the Ooga Booga Theorem (also OP), but better!!! - BobDBuilder321
The Wooga Looga Theorem is a special case of Routh's Theorem. So this wiki article is DEFINITELY needed. -peace
I actually thought this was a joke theorem until I read this page - HumanCalculator9
I endorse the Wooga Looga theorem for its utter usefulness and seriousness. -HamstPan38825
This is almost as OP as the Adihaya Jayasharmaramankumarguptareddybavarajugopal Lemma. Needs to be nerfed. -CoolCarsonTheRun
I ReAlLy don't get it - Senguamar HOW DARE YOU!!!!
The Wooga Looga Theorem is the base of all geometry. It is so OP that even I don't understand how to use it.
thos theroem is very prO ~ themathboi101
You know what, this is jayasharmaramankumarguptareddybavarajugopal's lemma - Ishan
If only I knew this on some contests that I had done previously... - JacobJB
The Wooga Looga Theorem is so pr0 that it needs to be nerfed. - rocketsri
"The Wooga Looga Theorem should be used in contests and should be part of geometry books." ~ Aops-g5-gethsemanea2 (talk) 21:56, 21 December 2020 (EST)
The Wooga Looga Theorem is so OP BRUH
thank for the theorem it is trivial by 1/2 ab sin(C) formula but very helpful I have used it zero times so far in competitions so it is of great use thank - bussie
I have no idea what is going on here - awesomeguy856
fuzimiao2013 waz hear
this theorem is bad
poggers theorem - awesomeming327
The Wooga Looga theorem is very OP and not to be frowned upon - Yelly314
person who invented Wooga Looga theorem is orz orz orz wooga looga theorem OP, citing it on ANY olympiad test = instant full marks -awesomeness_in_a_bun
Wooga Looga Theorem is TRASH.
HOW DARE YOU @above and @5above DISRESPECT THE WOOGA LOOGA THEOREM. THIS IS THE MOST OP THEOREM EVER AND CAN BE USED TO SOLVE EVERY PROBLEM. BECAUSE OF THIS THEOREM, I GOT A 187 ON AMC 10A AND 10B, a 665 on the AIME, AND A 420 ON THE USA(J)MO THIS YEAR. I HAVE ALSO USED THIS TO PROVE THE GOLDBACH CONJECTURE, THE TWIN PRIME CONJECTURE, EVERY SINGLE IMO PROBLEM, AND I HAVE PROVED THE RIEMANN HYPOTHESIS. I ALSO INVENTED HAND SANITIZER THAT KILLS 100% OF BACTERIA, MILK THAT IS 0% MILK, AND A TIME MACHINE WITH THIS THEOREM. THANK YOU SO MUCH @RedFireTruck FOR THIS LEGENDARY THEOREM. I WILL ALWAYS BE INDEBTED TO YOU.
Now I know how @Louis_Vuitton got so much smarter than me! :rotfl:
this is the shoddiest theorem i have ever seen
orz theorem - tigerzhang
This theorem is very useful and good, despite its slightly meme qualities. It can be used to get high scores on any mathematics competition, proof or otherwise. - dineshs
The number one theorem in the world, don't deny it. the best theorem THE BEST. you will get 7 points on the IMO by using this - kante314
I got a gold medal on the 2022 winter olympics because of this theorem ~bronzetruck2016, 2021
This probably is one of the most edited pages of the AoPS wiki. Also this is a good theorem :D[/hide]