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
Another solution involves using the proof of contradiction.
We first let a real number be the solution of the problem, which we set . By using induction, we can find out that such does not exist. Because the process of induction involves the use of calculus, fractals, and chaos theory, it is omitted in order to not make this page too long (to threaten the position of the great Gmass). However, you can find details of the proof here.
Since such does not exist, it is clearly a contradiction. The solution will be all real numbers that are not in the domain of , which is .
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
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.
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 USAJMO 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.