Wooga Looga Theorem
- 1 Definition
- 2 Proofs
- 3 Application 1
- 4 Application 2
- 5 Application 3
- 6 Application 4
- 7 Testimonials
If there is and points on the sides respectively such that , then the ratio .
Created by Foogle and Hoogle of The Ooga Booga Tribe of The Caveman Society
Proof by Gogobao:
So we have
Proof by franzliszt
Apply Barycentrics w.r.t. . Then . We can also find that . In the barycentric coordinate system, the area formula is where is a random triangle and is the reference triangle. Using this, we find that
WLOG we let , , for , . We then use Shoelace Forumla to get . We then figure out that , , and so we know that by Shoelace Formula . We know that for all so .
Proof by ishanvannadil2008:
Just use jayasharmaramankumarguptareddybavarajugopal's lemma. (Thanks to tenebrine)
The Wooga Looga Theorem states that the solution to this problem by franzliszt:
In points are on sides such that . Find the ratio of to .
WLOG let , , . Then by Shoelace Theorem and , , . Then by Shoelace Theorem. Therefore the answer is .
or this solution by franzliszt:
We apply Barycentric Coordinates w.r.t. . Let . Then we find that . In the barycentric coordinate system, the area formula is where is a random triangle and is the reference triangle. Using this, we find that
or this solution by aaja3427:
According the the Wooga Looga Theorem, It is . This is
or this solution by AoPS user ilovepizza2020:
We use the to instantly get . (Note: You can only use this method when you are not in a contest, as this method is so overpowered that the people behind mathematics examinations decided to ban it.)
or this solution by eduD_looC:
This is a perfect application of the Adihaya Jayasharmaramankumarguptareddybavarajugopal's Lemma, which results in the answer being . A very beautiful application, which leaves graders and readers speechless. Great for math contests with proofs.
or this solution by CoolJupiter:
Wow. All of your solutions are slow, compared to my sol:
By math, we have .
~CoolJupiter ^ | EVERYONE USE THIS SOLUTION IT'S BRILLIANT ~bsu1
Yes, very BRILLIANT!
The Best Solution
By the principle, we get . Definitely the best method. When asked, please say that OlympusHero taught you this method. Because he did.
The answer is clearly . We leave the proof and intermediate steps to the reader as an exercise.
The Wooga Looga Theorem states that the solution to this problem by Matholic:
The figure below shows a triangle ABC whose area is . If , find
~LaTeX-ifyed by RP3.1415
is this solution by franzliszt:
We apply Barycentric Coordinates w.r.t. . Let . Then we find that . In the barycentric coordinate system, the area formula is where is a random triangle and is the reference triangle. Using this, we find that so .
By the Wooga Looga Theorem, . We are given that so
The Wooga Looga Theorem states that the solution to this problem by RedFireTruck:
Find the ratio if and in the diagram below.
is this solution by franzliszt:
By the Wooga Looga Theorem, . Notice that is the medial triangle of Wooga Looga Triangle of . So and by Chain Rule ideas.
or this solution by franzliszt:
Apply Barycentrics w.r.t. so that . Then . And .
In the barycentric coordinate system, the area formula is where is a random triangle and is the reference triangle. Using this, we find that
Let be a triangle and be points on sides and respectively. We have that and similar for the other sides. If the area of triangle is , then what is the area of triangle ? (By ilovepizza2020)
By Wooga Looga, so the answer is .
Apply Barycentrics w.r.t. . Then . We can also find that . In the barycentric coordinate system, the area formula is where is a random triangle and is the reference triangle. Using this, we find thatSo the answer is .
A long story short, the answer must be by the inverse of the Inverse Wooga Looga Theorem
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 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 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