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| | "Very nice theorem" - [[User:RedFireTruck|<font color="#FF0000">RedFireTruck</font>]] ([[User talk:RedFireTruck|<font color="#FF0000">talk</font>]]) 12:12, 1 February 2021 (EST) | | "Very nice theorem" - [[User:RedFireTruck|<font color="#FF0000">RedFireTruck</font>]] ([[User talk:RedFireTruck|<font color="#FF0000">talk</font>]]) 12:12, 1 February 2021 (EST) |
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| − | [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
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| − | Thanks for rediscovering our theorem RedFireTruck - Foogle and Hoogle of The Ooga Booga Tribe of The Caveman Society
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| − | Franzlist is wooga looga howsopro - volkie boy
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| − | this is in fact a pretty sensible theorem. Nothing to be so excited about, though. ~DofL
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| − | The Wooga Looga Theorem is EPIC POGGERS WHOLESOME 100 KEANU CHUNGUS AMAZING SKILL THEOREM!!!!!1!!!111111 -centslordm
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| − | 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)
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| − | The Wooga Looga Theorem is the best. -aaja3427
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| − | The Wooga Looga Theorem is needed for everything and it is great-hi..
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| − | 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
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| − | This theorem has helped me with school and I am no longer failing my math class. -mchang
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| − | 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
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| − | "I can't believe AoPS books don't have this amazing theorem. If you need help with math, you can depend on caveman." ~CoolJupiter
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| − | 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)
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| − | Too powerful... ~franzliszt
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| − | The Wooga Looga Theorem is so pro ~ ac142931
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| − | It is so epic and awesome that it will blow the minds of people if they saw this ~ ac142931(2nd testimonial by me)
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| − | This theorem changed my life... ~ samrocksnature
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| − | Math competitions need to ban the use of the Wooga Looga Theorem, it's just too good. ~ jasperE3
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| − | It actually can be. I never thought I'd say this, but the Wooga Looga theorem is a legit theorem. ~ jasperE3
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| − | This is franzliszt and I endorse this theorem. ~franzliszt
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| − | This theorem is too OP. ~bestzack66
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| − | This is amazing! However much it looks like a joke, it is a legitimate - and powerful - theorem. -Supernova283
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| − | 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
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| − | The Wooga Looga cavemen were way ahead of their time. Good job (dead) guys! -HIA2020
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| − | It's like the Ooga Booga Theorem (also OP), but better!!! - BobDBuilder321
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| − | The Wooga Looga Theorem is a special case of Routh's Theorem. So this wiki article is DEFINITELY needed. -peace
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| − | I actually thought this was a joke theorem until I read this page - HumanCalculator9
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| − | I endorse the Wooga Looga theorem for its utter usefulness and seriousness. -HamstPan38825
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| − | This is almost as OP as the Adihaya Jayasharmaramankumarguptareddybavarajugopal Lemma. Needs to be nerfed. -CoolCarsonTheRun
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| − | I ReAlLy don't get it - Senguamar HOW DARE YOU!!!!
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| − | The Wooga Looga Theorem is the base of all geometry. It is so OP that even I don't understand how to use it.
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| − | thos theroem is very prO ~ themathboi101
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| − | You know what, this is jayasharmaramankumarguptareddybavarajugopal's lemma - Ishan
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| − | If only I knew this on some contests that I had done previously... - JacobJB
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| − | The Wooga Looga Theorem is so pr0 that it needs to be nerfed. - rocketsri
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| − | "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)
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| − | The Wooga Looga Theorem is so OP BRUH
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| − | 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
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| − | I have no idea what is going on here - awesomeguy856
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| − | fuzimiao2013 waz hear
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| − | this theorem is bad
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| − | poggers theorem - awesomeming327
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| − | The Wooga Looga theorem is very OP and not to be frowned upon - Yelly314
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| − | 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
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| − | Wooga Looga Theorem is TRASH.
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| − | 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.
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| − | Now I know how @Louis_Vuitton got so much smarter than me! :rotfl:
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| − | this is the shoddiest theorem i have ever seen
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| − | orz theorem - tigerzhang
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| − | 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
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| − | 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
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| − | I got a gold medal on the 2022 winter olympics because of this theorem ~bronzetruck2016, 2021
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| − | This probably is one of the most edited pages of the AoPS wiki. Also this is a good theorem :D[/hide]
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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 ![$\frac{[DEF]}{[ABC]}=1-\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}=\frac{rst+1}{(r+1)(s+1)(t+1)}$](//latex.artofproblemsolving.com/2/b/3/2b3e9d3f239c1b1129cebf63683811855f732540.png)
.
(*Simplification found by @Gogobao)
Proofs
Proof 1
Proof by CoolJupiter:
We have the following ratios:
.
Now notice that ![$[DEF]=[ABC]-([BDF]+[CDE]+[AEF])$](//latex.artofproblemsolving.com/3/4/8/348cf5258a141132df51ae2e1ec60dd89e37d47f.png)
.
We attempt to find the area of each of the smaller triangles.
Notice that ![$\frac{[BDF]}{[ABC]}=\frac{BF}{AB}\times \frac{BD}{BC}=\frac{r}{(r+1)(t+1)}$](//latex.artofproblemsolving.com/e/1/f/e1f4ef80cad3ed00c9d0bc6fb8b2f56ed0faca5f.png)
using the ratios derived earlier.
Similarly, ![$\frac{[CDE]}{[ABC]}=\frac{s}{(r+1)(s+1)}$](//latex.artofproblemsolving.com/6/a/d/6ad09b527364466a3f92f10e3170e1600f3fd300.png)
and ![$\frac{[AEF]}{[ABC]}=\frac{t}{(s+1)(t+1)}$](//latex.artofproblemsolving.com/f/b/d/fbd5231991313efc841a68b5bcbb6794dfec3004.png)
.
Thus, ![$\frac{[BDF]+[CDE]+[AEF]}{[ABC]}=\frac{r}{(r+1)(t+1)}+\frac{s}{(r+1)(s+1)}+\frac{t}{(s+1)(t+1)}=\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}$](//latex.artofproblemsolving.com/a/0/1/a012303031a664b7d67454d68942d25743528b37.png)
.
Finally, we have ![$\frac{[DEF]}{[ABC]}=1-\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}=\boxed{\frac{rst+1}{(r+1)(s+1)(t+1)}}$](//latex.artofproblemsolving.com/a/6/0/a60005eb968b9cb4f8d5777ec9abf0e2d0cd85fb.png)
.
~@CoolJupiter
Proof 2
Proof by math_comb01
Apply Barycentrics . Then 
. also 
In the barycentrics, the area formula is ![$[XYZ]=\begin{vmatrix} x_{1} &y_{1} &z_{1} \\ x_{2} &y_{2} &z_{2} \\ x_{3}& y_{3} & z_{3} \end{vmatrix}\cdot [ABC]$](//latex.artofproblemsolving.com/e/d/f/edf87c7f6a6293bef13c56e0097c45996302e580.png)
where 
is a random triangle and is the reference triangle. Using this, we ![\[\frac{[DEF]}{[ABC]}\]](//latex.artofproblemsolving.com/9/9/6/99648334a37fce73b84b7a8267d611cffd163ada.png)
=
=![$\frac{1}{[s+1][r+1][t+1]}$](//latex.artofproblemsolving.com/f/3/1/f31eed443729b8cb339df8c46be103bf2d77cda2.png)
![$+\frac{rst}{([s+1][r+1][t+1]}$](//latex.artofproblemsolving.com/f/e/d/fed79961244d3ddab2d9e6cbe9e3cd77fc2a4c70.png)
=![$\frac{rst+1}{([s+1][r+1][t+1]}$](//latex.artofproblemsolving.com/8/7/8/8780566469d3a864d36aa883a8a04c4e68e4808a.png)
~@Math_comb01
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)
, let 
and 
be points on 
and 
respectively. The Generalized Wooga Looga Theorem (Gwoologth) or the Devil's Triangle Theorem states that if 
and 
, then ![$\frac{[DEF]}{[ABC]}=1-\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}=\frac{rst+1}{(r+1)(s+1)(t+1)}$](http://latex.artofproblemsolving.com/2/b/3/2b3e9d3f239c1b1129cebf63683811855f732540.png)
.
.
![$[DEF]=[ABC]-([BDF]+[CDE]+[AEF])$](http://latex.artofproblemsolving.com/3/4/8/348cf5258a141132df51ae2e1ec60dd89e37d47f.png)
.
![$\frac{[BDF]}{[ABC]}=\frac{BF}{AB}\times \frac{BD}{BC}=\frac{r}{(r+1)(t+1)}$](http://latex.artofproblemsolving.com/e/1/f/e1f4ef80cad3ed00c9d0bc6fb8b2f56ed0faca5f.png)
using the ratios derived earlier.
![$\frac{[CDE]}{[ABC]}=\frac{s}{(r+1)(s+1)}$](http://latex.artofproblemsolving.com/6/a/d/6ad09b527364466a3f92f10e3170e1600f3fd300.png)
and ![$\frac{[AEF]}{[ABC]}=\frac{t}{(s+1)(t+1)}$](http://latex.artofproblemsolving.com/f/b/d/fbd5231991313efc841a68b5bcbb6794dfec3004.png)
.
![$\frac{[BDF]+[CDE]+[AEF]}{[ABC]}=\frac{r}{(r+1)(t+1)}+\frac{s}{(r+1)(s+1)}+\frac{t}{(s+1)(t+1)}=\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}$](http://latex.artofproblemsolving.com/a/0/1/a012303031a664b7d67454d68942d25743528b37.png)
.
![$\frac{[DEF]}{[ABC]}=1-\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}=\boxed{\frac{rst+1}{(r+1)(s+1)(t+1)}}$](http://latex.artofproblemsolving.com/a/6/0/a60005eb968b9cb4f8d5777ec9abf0e2d0cd85fb.png)
.
. also 
![$[XYZ]=\begin{vmatrix} x_{1} &y_{1} &z_{1} \\ x_{2} &y_{2} &z_{2} \\ x_{3}& y_{3} & z_{3} \end{vmatrix}\cdot [ABC]$](http://latex.artofproblemsolving.com/e/d/f/edf87c7f6a6293bef13c56e0097c45996302e580.png)
where 
is a random triangle and ![\[\frac{[DEF]}{[ABC]}\]](http://latex.artofproblemsolving.com/9/9/6/99648334a37fce73b84b7a8267d611cffd163ada.png)
=
=![$\frac{1}{[s+1][r+1][t+1]}$](http://latex.artofproblemsolving.com/f/3/1/f31eed443729b8cb339df8c46be103bf2d77cda2.png)
![$+\frac{rst}{([s+1][r+1][t+1]}$](http://latex.artofproblemsolving.com/f/e/d/fed79961244d3ddab2d9e6cbe9e3cd77fc2a4c70.png)
=![$\frac{rst+1}{([s+1][r+1][t+1]}$](http://latex.artofproblemsolving.com/8/7/8/8780566469d3a864d36aa883a8a04c4e68e4808a.png)
.
but ok ~ bissue
