# Difference between revisions of "Wooga Looga Theorem"

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− | This | + | =Definition= |

+ | If there is <math>\triangle ABC</math> and points <math>D,E,F</math> on the sides <math>BC,CA,AB</math> respectively such that <math>\frac{DB}{DC}=\frac{EC}{EA}=\frac{FA}{FB}=r</math>, then the ratio <math>\frac{[DEF]}{[ABC]}=\frac{r^2-r+1}{(r+1)^2}</math>. | ||

+ | |||

+ | |||

+ | Created by Foogle and Hoogle of [https://www.youtube.com/channel/UC50E9TuLIMWbOPUX45xZPaQ The Ooga Booga Tribe of The Caveman Society] | ||

+ | |||

+ | =Proofs= | ||

+ | ==Proof 1== | ||

+ | Proof by Gogobao: | ||

+ | |||

+ | We have: <math>\frac{DB}{BC} = \frac{r}{r+1}, \frac{DC}{BC} = \frac{1}{r+1}, \frac{EC}{AC} = \frac{r}{r+1}, \frac{EA}{AC} = \frac{1}{r+1}, \frac{FA}{BA} = \frac{r}{r+1}, \frac{FB}{BA} = \frac{1}{r+1} </math> | ||

+ | |||

+ | We have: <math>[DEF] = [ABC] - [DCE] - [FAE] - [FBD]</math> | ||

+ | |||

+ | <math>[DCE] = [ABC] \cdot \frac{DC}{CB} \cdot \frac{CE}{CA} = [ABC] \cdot \frac{r}{(r+1)^2}</math> | ||

+ | |||

+ | <math>[EAF] = [ABC] \cdot \frac{EA}{CA} \cdot \frac{FA}{BA} = [ABC] \cdot \frac{r}{(r+1)^2}</math> | ||

+ | |||

+ | <math>[FBD] = [ABC] \cdot \frac{FB}{AB} \cdot \frac{BD}{CB} = [ABC] \cdot \frac{r}{(r+1)^2}</math> | ||

+ | |||

+ | Therefore <math>[DEF] = [ABC] (1-\frac{3r}{(r+1)^2})</math> | ||

+ | |||

+ | So we have <math>\frac{[DEF]}{[ABC]} = \frac{r^2-r+1}{(r+1)^2}</math> | ||

+ | |||

+ | ==Proof 2== | ||

+ | Proof by franzliszt | ||

+ | |||

+ | Apply Barycentrics w.r.t. <math>\triangle ABC</math>. Then <math>A=(1,0,0),B=(0,1,0),C=(0,0,1)</math>. We can also find that <math>D=\left(0,\tfrac {1}{r+1},\tfrac {r}{r+1}\right),E=\left(\tfrac {r}{r+1},0,\tfrac {1}{r+1}\right),F=\left(\tfrac {1}{r+1},\tfrac {r}{r+1},0\right)</math>. In the barycentric coordinate system, the area formula is <math>[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]</math> where <math>\triangle XYZ</math> is a random triangle and <math>\triangle ABC</math> is the reference triangle. Using this, we find that <cmath>\frac{[DEF]}{[ABC]}= \begin{vmatrix} 0&\tfrac {1}{r+1}&\tfrac {r}{r+1} \\ \tfrac {r}{r+1}&0&\tfrac {1}{r+1}\\ \tfrac {1}{r+1}&\tfrac {r}{r+1}&0 \end{vmatrix}=\frac{r^2-r+1}{(r+1)^2}.</cmath> | ||

+ | |||

+ | ==Proof 3== | ||

+ | Proof by [[User:RedFireTruck|<font color="#FF0000">RedFireTruck</font>]] ([[User talk:RedFireTruck|<font color="#FF0000">talk</font>]]) 12:11, 1 February 2021 (EST): | ||

+ | |||

+ | WLOG we let <math>A=(0, 0)</math>, <math>B=(1, 0)</math>, <math>C=(x, y)</math> for <math>x</math>, <math>y\in\mathbb{R}</math>. We then use Shoelace Forumla to get <math>[ABC]=\frac12|y|</math>. We then figure out that <math>D=\left(\frac{rx+1}{r+1}, \frac{ry}{r+1}\right)</math>, <math>E=\left(\frac{x}{r+1}, \frac{y}{r+1}\right)</math>, and <math>F=\left(\frac{r}{r+1}, 0\right)</math> so we know that by Shoelace Formula <math>\frac{[DEF]}{[ABC]}=\frac{\frac12\left|\frac{r^2y-ry+y}{(r+1)^2}\right|}{\frac12|y|}=\left|\frac{r^2-r+1}{(r+1)^2}\right|</math>. We know that <math>\frac{r^2-r+1}{(r+1)^2}\ge0</math> for all <math>r\in\mathbb{R}</math> so <math>\left|\frac{r^2-r+1}{(r+1)^2}\right|=\frac{r^2-r+1}{(r+1)^2}</math>. | ||

+ | |||

+ | ==Proof 4== | ||

+ | Proof by ishanvannadil2008: | ||

+ | |||

+ | Just use jayasharmaramankumarguptareddybavarajugopal's lemma. (Thanks to tenebrine) | ||

+ | |||

+ | =Application 1= | ||

+ | ==Problem== | ||

+ | The Wooga Looga Theorem states that the solution to this problem by franzliszt: | ||

+ | |||

+ | In <math>\triangle ABC</math> points <math>X,Y,Z</math> are on sides <math>BC,CA,AB</math> such that <math>\frac{XB}{XC}=\frac{YC}{YA}=\frac{ZA}{ZB}=\frac 71</math>. Find the ratio of <math>[XYZ]</math> to <math>[ABC]</math>. | ||

+ | |||

+ | ==Solution 1== | ||

+ | One solution is this one by [[User:RedFireTruck|<font color="#FF0000">RedFireTruck</font>]] ([[User talk:RedFireTruck|<font color="#FF0000">talk</font>]]) 12:11, 1 February 2021 (EST): | ||

+ | |||

+ | WLOG let <math>A=(0, 0)</math>, <math>B=(1, 0)</math>, <math>C=(x, y)</math>. Then <math>[ABC]=\frac12|y|</math> by Shoelace Theorem and <math>X=\left(\frac{7x+1}{8}, \frac{7y}{8}\right)</math>, <math>Y=\left(\frac{x}{8}, \frac{y}{8}\right)</math>, <math>Z=\left(\frac78, 0\right)</math>. Then <math>[XYZ]=\frac12\left|\frac{43y}{64}\right|</math> by Shoelace Theorem. Therefore the answer is <math>\boxed{\frac{43}{64}}</math>. | ||

+ | |||

+ | ==Solution 2== | ||

+ | or this solution by franzliszt: | ||

+ | |||

+ | We apply Barycentric Coordinates w.r.t. <math>\triangle ABC</math>. Let <math>A=(1,0,0),B=(0,1,0),C=(0,0,1)</math>. Then we find that <math>X=\left(0,\tfrac 18,\tfrac 78\right),Y=\left(\tfrac 78,0,\tfrac 18\right),Z=\left(\tfrac18,\tfrac78,0\right)</math>. In the barycentric coordinate system, the area formula is <math>[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]</math> where <math>\triangle XYZ</math> is a random triangle and <math>\triangle ABC</math> is the reference triangle. Using this, we find that <cmath>\frac{[XYZ]}{[ABC]}=\begin{vmatrix} | ||

+ | 0&\tfrac 18&\tfrac 78\\ | ||

+ | \tfrac 78&0&\tfrac 18\\ | ||

+ | \tfrac18&\tfrac78&0 | ||

+ | \end{vmatrix}=\frac{43}{64}.</cmath> <math>\blacksquare</math> | ||

+ | ==Solution 3== | ||

+ | or this solution by aaja3427: | ||

+ | |||

+ | According the the Wooga Looga Theorem, It is <math>\frac{49-7+1}{8^2}</math>. This is <math>\boxed{\frac{43}{64}}</math> | ||

+ | |||

+ | ==Solution 4== | ||

+ | |||

+ | or this solution by AoPS user ilovepizza2020: | ||

+ | |||

+ | We use the <math>\mathbf{FUNDAMENTAL~THEOREM~OF~GEOGEBRA}</math> to instantly get <math>\boxed{\frac{43}{64}}</math>. (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.) | ||

+ | |||

+ | ==Solution 5== | ||

+ | or this solution by eduD_looC: | ||

+ | |||

+ | This is a perfect application of the Adihaya Jayasharmaramankumarguptareddybavarajugopal's Lemma, which results in the answer being <math>\boxed{\frac{43}{64}}</math>. A very beautiful application, which leaves graders and readers speechless. Great for math contests with proofs. | ||

+ | |||

+ | ==Solution 6== | ||

+ | or this solution by CoolJupiter: | ||

+ | |||

+ | Wow. All of your solutions are slow, compared to my sol: | ||

+ | |||

+ | By math, we have <math>\boxed{\frac{43}{64}}</math>. | ||

+ | |||

+ | ~CoolJupiter | ||

+ | ^ | ||

+ | | | ||

+ | EVERYONE USE THIS SOLUTION IT'S BRILLIANT | ||

+ | ~bsu1 | ||

+ | Yes, very BRILLIANT! | ||

+ | ~ TheAoPSLebron | ||

+ | |||

+ | ==The Best Solution== | ||

+ | |||

+ | By the <math>1+1=\text{BREAD}</math> principle, we get <math>\boxed{\frac{43}{64}}</math>. Definitely the best method. When asked, please say that OlympusHero taught you this method. Because he did. | ||

+ | |||

+ | ==Easiest Solution== | ||

+ | |||

+ | The answer is clearly <math>\boxed{\frac{43}{64}}</math>. We leave the proof and intermediate steps to the reader as an exercise. | ||

+ | |||

+ | =Application 2= | ||

+ | ==Problem== | ||

+ | The Wooga Looga Theorem states that the solution to this problem by Matholic: | ||

+ | |||

+ | The figure below shows a triangle ABC whose area is <math>72 \text{cm}^2</math>. If <math>\dfrac{AD}{DB}=\dfrac{BE}{EC}=\dfrac{CF}{FA}=\dfrac{1}{5}</math>, find <math>[DEF].</math> | ||

+ | |||

+ | ~LaTeX-ifyed by RP3.1415 | ||

+ | |||

+ | ==Solution 1== | ||

+ | is this solution by franzliszt: | ||

+ | |||

+ | We apply Barycentric Coordinates w.r.t. <math>\triangle ABC</math>. Let <math>A=(1,0,0),B=(0,1,0),C=(0,0,1)</math>. Then we find that <math>D=(\tfrac 56,\tfrac 16,0),E=(0,\tfrac 56,\tfrac 16),F=(\tfrac16,0,\tfrac56)</math>. In the barycentric coordinate system, the area formula is <math>[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]</math> where <math>\triangle XYZ</math> is a random triangle and <math>\triangle ABC</math> is the reference triangle. Using this, we find that<cmath>\frac{[DEF]}{[72]}=\begin{vmatrix} | ||

+ | \tfrac 56&\tfrac 16&0\\ | ||

+ | 0&\tfrac 56&\tfrac 16\\ | ||

+ | \tfrac16&0&\tfrac56 | ||

+ | \end{vmatrix}=\frac{7}{12}</cmath> so <math>[DEF]=42</math>. <math>\blacksquare</math> | ||

+ | ==Solution 2== | ||

+ | or this solution by [[User:RedFireTruck|<font color="#FF0000">RedFireTruck</font>]] ([[User talk:RedFireTruck|<font color="#FF0000">talk</font>]]) 12:11, 1 February 2021 (EST): | ||

+ | |||

+ | By the Wooga Looga Theorem, <math>\frac{[DEF]}{[ABC]}=\frac{5^2-5+1}{(5+1)^2}=\frac{21}{36}=\frac{7}{12}</math>. We are given that <math>[ABC]=72</math> so <math>[DEF]=\frac{7}{12}\cdot72=\boxed{42}</math> | ||

+ | |||

+ | =Application 3= | ||

+ | ==Problem== | ||

+ | The Wooga Looga Theorem states that the solution to this problem by RedFireTruck: | ||

+ | |||

+ | Find the ratio <math>\frac{[GHI]}{[ABC]}</math> if <math>\frac{AD}{DB}=\frac{BE}{EC}=\frac{CF}{FA}=\frac12</math> and <math>\frac{DG}{GE}=\frac{EH}{HF}=\frac{FI}{ID}=1</math> in the diagram below.<asy> | ||

+ | draw((0, 0)--(6, 0)--(4, 3)--cycle); | ||

+ | draw((2, 0)--(16/3, 1)--(8/3, 2)--cycle); | ||

+ | draw((11/3, 1/2)--(4, 3/2)--(7/3, 1)--cycle); | ||

+ | label("$A$", (0, 0), SW); | ||

+ | label("$B$", (6, 0), SE); | ||

+ | label("$C$", (4, 3), N); | ||

+ | label("$D$", (2, 0), S); | ||

+ | label("$E$", (16/3, 1), NE); | ||

+ | label("$F$", (8/3, 2), NW); | ||

+ | label("$G$", (11/3, 1/2), SE); | ||

+ | label("$H$", (4, 3/2), NE); | ||

+ | label("$I$", (7/3, 1), W); | ||

+ | </asy> | ||

+ | ==Solution 1== | ||

+ | is this solution by franzliszt: | ||

+ | |||

+ | By the Wooga Looga Theorem, <math>\frac{[DEF]}{[ABC]}=\frac{2^2-2+1}{(1+2)^2}=\frac 13</math>. Notice that <math>\triangle GHI</math> is the medial triangle of '''Wooga Looga Triangle ''' of <math>\triangle ABC</math>. So <math>\frac{[GHI]}{[DEF]}=\frac 14</math> and <math>\frac{[GHI]}{[ABD]}=\frac{[DEF]}{[ABC]}\cdot\frac{[GHI]}{[DEF]}=\frac 13 \cdot \frac 14 = \frac {1}{12}</math> by Chain Rule ideas. | ||

+ | |||

+ | ==Solution 2== | ||

+ | or this solution by franzliszt: | ||

+ | |||

+ | Apply Barycentrics w.r.t. <math>\triangle ABC</math> so that <math>A=(1,0,0),B=(0,1,0),C=(0,0,1)</math>. Then <math>D=(\tfrac 23,\tfrac 13,0),E=(0,\tfrac 23,\tfrac 13),F=(\tfrac 13,0,\tfrac 23)</math>. And <math>G=(\tfrac 13,\tfrac 12,\tfrac 16),H=(\tfrac 16,\tfrac 13,\tfrac 12),I=(\tfrac 12,\tfrac 16,\tfrac 13)</math>. | ||

+ | |||

+ | In the barycentric coordinate system, the area formula is <math>[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]</math> where <math>\triangle XYZ</math> is a random triangle and <math>\triangle ABC</math> is the reference triangle. Using this, we find that<cmath>\frac{[GHI]}{[ABC]}=\begin{vmatrix} \tfrac 13&\tfrac 12&\tfrac 16\\ \tfrac 16&\tfrac 13&\tfrac 12\\ \tfrac 12&\tfrac 16&\tfrac 13 \end{vmatrix}=\frac{1}{12}.</cmath> | ||

+ | |||

+ | =Application 4= | ||

+ | ==Problem== | ||

+ | |||

+ | Let <math>ABC</math> be a triangle and <math>D,E,F</math> be points on sides <math>BC,AC,</math> and <math>AB</math> respectively. We have that <math>\frac{BD}{DC} = 3</math> and similar for the other sides. If the area of triangle <math>ABC</math> is <math>16</math>, then what is the area of triangle <math>DEF</math>? (By ilovepizza2020) | ||

+ | |||

+ | ==Solution 1== | ||

+ | |||

+ | By Franzliszt | ||

+ | |||

+ | By Wooga Looga, <math>\frac{[DEF]}{16} = \frac{3^2-3+1}{(3+1)^2}=\frac{7}{16}</math> so the answer is <math>\boxed7</math>. | ||

+ | |||

+ | ==Solution 2== | ||

+ | |||

+ | By franzliszt | ||

+ | |||

+ | Apply Barycentrics w.r.t. <math>\triangle ABC</math>. Then <math>A=(1,0,0),B=(0,1,0),C=(0,0,1)</math>. We can also find that <math>D=(0,\tfrac 14,\tfrac 34),E=(\tfrac 34,0,\tfrac 14),F=(\tfrac 14,\tfrac 34,0)</math>. In the barycentric coordinate system, the area formula is <math>[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]</math> where <math>\triangle XYZ</math> is a random triangle and <math>\triangle ABC</math> is the reference triangle. Using this, we find that<cmath>\frac{[DEF]}{[ABC]}=\begin{vmatrix} 0&\tfrac 14&\tfrac 34\\ \tfrac 34&0&\tfrac 14\\ \tfrac 14&\tfrac 34&0 \end{vmatrix}=\frac{7}{16}.</cmath>So the answer is <math>\boxed{7}</math>. | ||

+ | |||

+ | ==Solution 3== | ||

+ | |||

+ | A long story short, the answer must be <math>\boxed{7}</math> by the inverse of the Inverse Wooga Looga Theorem | ||

+ | |||

+ | =Testimonials= | ||

+ | |||

+ | 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 [https://www.youtube.com/channel/UC50E9TuLIMWbOPUX45xZPaQ 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. - [[User:RedFireTruck|<font color="#FF0000">RedFireTruck</font>]] ([[User talk:RedFireTruck|<font color="#FF0000">talk</font>]]) 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 [https://en.wikipedia.org/wiki/Routh%27s_theorem 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 | ||

+ | |||

+ | <s>I ReAlLy don't get it - Senguamar</s> 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 should be used in contests and should be part of geometry books." ~ [[User:Aops-g5-gethsemanea2|Aops-g5-gethsemanea2]] ([[User talk: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 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. |

## Revision as of 23:00, 26 April 2021

## Contents

# Definition

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

# Proofs

## Proof 1

Proof by Gogobao:

We have:

We have:

Therefore

So we have

## Proof 2

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

## Proof 3

Proof by RedFireTruck (talk) 12:11, 1 February 2021 (EST):

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 4

Proof by ishanvannadil2008:

Just use jayasharmaramankumarguptareddybavarajugopal's lemma. (Thanks to tenebrine)

# Application 1

## Problem

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 .

## Solution 1

One solution is this one by RedFireTruck (talk) 12:11, 1 February 2021 (EST):

WLOG let , , . Then by Shoelace Theorem and , , . Then by Shoelace Theorem. Therefore the answer is .

## Solution 2

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

## Solution 3

or this solution by aaja3427:

According the the Wooga Looga Theorem, It is . This is

## Solution 4

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.)

## Solution 5

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.

## Solution 6

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!

~ TheAoPSLebron

## 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.

## Easiest Solution

The answer is clearly . We leave the proof and intermediate steps to the reader as an exercise.

# Application 2

## Problem

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

## Solution 1

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 .

## Solution 2

or this solution by RedFireTruck (talk) 12:11, 1 February 2021 (EST):

By the Wooga Looga Theorem, . We are given that so

# Application 3

## Problem

The Wooga Looga Theorem states that the solution to this problem by RedFireTruck:

Find the ratio if and in the diagram below.

## Solution 1

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.

## Solution 2

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

# Application 4

## Problem

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)

## Solution 1

By Franzliszt

By Wooga Looga, so the answer is .

## Solution 2

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 thatSo the answer is .

## Solution 3

A long story short, the answer must be by the inverse of the Inverse Wooga Looga Theorem

# Testimonials

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

~~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 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 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.