Latest revision as of 17:05, 23 March 2024

Definition

Generalized Wooga Looga Theorem (The Devil's Triangle)

For any triangle $\triangle ABC$ , let $D, E$ and $F$ be points on $BC, AC$ and $AB$ respectively. The Generalized Wooga Looga Theorem (Gwoologth) or the Devil's Triangle Theorem states that if $\frac{BD}{CD}=r, \frac{CE}{AE}=s$ and $\frac{AF}{BF}=t$ , then $\frac{[DEF]}{[ABC]}=1-\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}=\frac{rst+1}{(r+1)(s+1)(t+1)}$ .

(*Simplification found by @Gogobao)

Proofs

Proof 1

Proof by CoolJupiter:

We have the following ratios: $\frac{BD}{BC}=\frac{r}{r+1}, \frac{CD}{BC}=\frac{1}{r+1},\frac{CE}{AC}=\frac{s}{s+1}, \frac{AE}{AC}=\frac{1}{s+1},\frac{AF}{AB}=\frac{t}{t+1}, \frac{BF}{AB}=\frac{1}{t+1}$ .

Now notice that $[DEF]=[ABC]-([BDF]+[CDE]+[AEF])$ .

We attempt to find the area of each of the smaller triangles.

Notice that $\frac{[BDF]}{[ABC]}=\frac{BF}{AB}\times \frac{BD}{BC}=\frac{r}{(r+1)(t+1)}$ using the ratios derived earlier.

Similarly, $\frac{[CDE]}{[ABC]}=\frac{s}{(r+1)(s+1)}$ and $\frac{[AEF]}{[ABC]}=\frac{t}{(s+1)(t+1)}$ .

Thus, $\frac{[BDF]+[CDE]+[AEF]}{[ABC]}=\frac{r}{(r+1)(t+1)}+\frac{s}{(r+1)(s+1)}+\frac{t}{(s+1)(t+1)}=\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}$ .

Finally, we have $\frac{[DEF]}{[ABC]}=1-\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}=\boxed{\frac{rst+1}{(r+1)(s+1)(t+1)}}$ .

~@CoolJupiter

Proof 2

Proof by math_comb01 Apply Barycentrics $\triangle ABC$ . Then $A=(1,0,0),B=(0,1,0),C=(0,0,1)$ . also $D=\left(0,\tfrac {1}{r+1},\tfrac {r}{r+1}\right),E=\left(\tfrac {s}{s+1},0,\tfrac {1}{s+1}\right),F=\left(\tfrac {1}{t+1},\tfrac {t}{t+1},0\right)$

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://webcache.googleusercontent.com/search?q=cache:Qoyk2gGO6x8J:https://artofproblemsolving.com/wiki/index.php/Wooga_Looga_Theorem+&cd=1&hl=en&ct=clnk&gl=us&client=safari

Essentially, Wooga Looga is a special case of this, specifically when $r=s=t$ .

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 $\frac{1}{2}ab\sin(C)$ but ok ~ bissue

"Very nice theorem" - RedFireTruck (talk) 12:12, 1 February 2021 (EST)

“I in the o’l days I used this theorem all the when time trying to tame my mammoth my cave buddy told me to ooga booga” - peelybonehead 9,000 B.C.

who else got redirected here from 2004 AMC 10B Problem 18 smh

I love the Wooga Looga Theorem! ~ Math-lover1

@@ Line 36: / Line 36: @@
 =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
+https://webcache.googleusercontent.com/search?q=cache:Qoyk2gGO6x8J:https://artofproblemsolving.com/wiki/index.php/Wooga_Looga_Theorem+&cd=1&hl=en&ct=clnk&gl=us&client=safari
 Essentially, Wooga Looga is a special case of this, specifically when <math>r=s=t</math>.
@@ Line 50: / Line 51: @@
 "Very nice theorem" - [[User:RedFireTruck|<font color="#FF0000">RedFireTruck</font>]] ([[User talk:RedFireTruck|<font color="#FF0000">talk</font>]]) 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
+“I in the o’l days I used this theorem all the when time trying to tame my mammoth my cave buddy told me to ooga booga” - peelybonehead 9,000 B.C.
-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
+who else got redirected here from 2004 AMC 10B Problem 18 smh
-This probably is one of the most edited pages of the AoPS wiki. Also this is a good theorem :D[/hide]
+I love the Wooga Looga Theorem! ~ Math-lover1

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