https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Nodens&feedformat=atom AoPS Wiki - User contributions [en] 2021-06-22T02:31:36Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=Wooga_Looga_Theorem&diff=154289 Wooga Looga Theorem 2021-05-27T00:19:45Z <p>Nodens: </p> <hr /> <div>=Definition=<br /> If there is &lt;math&gt;\triangle ABC&lt;/math&gt; and points &lt;math&gt;D,E,F&lt;/math&gt; on the sides &lt;math&gt;BC,CA,AB&lt;/math&gt; respectively such that &lt;math&gt;\frac{DB}{DC}=\frac{EC}{EA}=\frac{FA}{FB}=r&lt;/math&gt;, then the ratio &lt;math&gt;\frac{[DEF]}{[ABC]}=\frac{r^2-r+1}{(r+1)^2}&lt;/math&gt;.<br /> <br /> <br /> Created by Foogle and Hoogle of [https://www.youtube.com/channel/UC50E9TuLIMWbOPUX45xZPaQ The Ooga Booga Tribe of The Caveman Society]<br /> <br /> =Proofs=<br /> ==Proof 1==<br /> Proof by Gogobao:<br /> <br /> We have: &lt;math&gt;\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} &lt;/math&gt;<br /> <br /> We have: &lt;math&gt;[DEF] = [ABC] - [DCE] - [FAE] - [FBD]&lt;/math&gt;<br /> <br /> &lt;math&gt;[DCE] = [ABC] \cdot \frac{DC}{CB} \cdot \frac{CE}{CA} = [ABC] \cdot \frac{r}{(r+1)^2}&lt;/math&gt;<br /> <br /> &lt;math&gt;[EAF] = [ABC] \cdot \frac{EA}{CA} \cdot \frac{FA}{BA} = [ABC] \cdot \frac{r}{(r+1)^2}&lt;/math&gt;<br /> <br /> &lt;math&gt;[FBD] = [ABC] \cdot \frac{FB}{AB} \cdot \frac{BD}{CB} = [ABC] \cdot \frac{r}{(r+1)^2}&lt;/math&gt;<br /> <br /> Therefore &lt;math&gt;[DEF] = [ABC] (1-\frac{3r}{(r+1)^2})&lt;/math&gt;<br /> <br /> So we have &lt;math&gt;\frac{[DEF]}{[ABC]} = \frac{r^2-r+1}{(r+1)^2}&lt;/math&gt;<br /> <br /> ==Proof 2==<br /> Proof by franzliszt<br /> <br /> Apply Barycentrics w.r.t. &lt;math&gt;\triangle ABC&lt;/math&gt;. Then &lt;math&gt;A=(1,0,0),B=(0,1,0),C=(0,0,1)&lt;/math&gt;. We can also find that &lt;math&gt;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)&lt;/math&gt;. In the barycentric coordinate system, the area formula is &lt;math&gt;[XYZ]=\begin{vmatrix} x_{1} &amp;y_{1} &amp;z_{1} \\ x_{2} &amp;y_{2} &amp;z_{2} \\ x_{3}&amp; y_{3} &amp; z_{3} \end{vmatrix}\cdot [ABC]&lt;/math&gt; where &lt;math&gt;\triangle XYZ&lt;/math&gt; is a random triangle and &lt;math&gt;\triangle ABC&lt;/math&gt; is the reference triangle. Using this, we find that &lt;cmath&gt;\frac{[DEF]}{[ABC]}= \begin{vmatrix} 0&amp;\tfrac {1}{r+1}&amp;\tfrac {r}{r+1} \\ \tfrac {r}{r+1}&amp;0&amp;\tfrac {1}{r+1}\\ \tfrac {1}{r+1}&amp;\tfrac {r}{r+1}&amp;0 \end{vmatrix}=\frac{r^2-r+1}{(r+1)^2}.&lt;/cmath&gt;<br /> <br /> ==Proof 3==<br /> Proof by [[User:RedFireTruck|&lt;font color=&quot;#FF0000&quot;&gt;RedFireTruck&lt;/font&gt;]] ([[User talk:RedFireTruck|&lt;font color=&quot;#FF0000&quot;&gt;talk&lt;/font&gt;]]) 12:11, 1 February 2021 (EST):<br /> <br /> WLOG we let &lt;math&gt;A=(0, 0)&lt;/math&gt;, &lt;math&gt;B=(1, 0)&lt;/math&gt;, &lt;math&gt;C=(x, y)&lt;/math&gt; for &lt;math&gt;x&lt;/math&gt;, &lt;math&gt;y\in\mathbb{R}&lt;/math&gt;. We then use Shoelace Forumla to get &lt;math&gt;[ABC]=\frac12|y|&lt;/math&gt;. We then figure out that &lt;math&gt;D=\left(\frac{rx+1}{r+1}, \frac{ry}{r+1}\right)&lt;/math&gt;, &lt;math&gt;E=\left(\frac{x}{r+1}, \frac{y}{r+1}\right)&lt;/math&gt;, and &lt;math&gt;F=\left(\frac{r}{r+1}, 0\right)&lt;/math&gt; so we know that by Shoelace Formula &lt;math&gt;\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|&lt;/math&gt;. We know that &lt;math&gt;\frac{r^2-r+1}{(r+1)^2}\ge0&lt;/math&gt; for all &lt;math&gt;r\in\mathbb{R}&lt;/math&gt; so &lt;math&gt;\left|\frac{r^2-r+1}{(r+1)^2}\right|=\frac{r^2-r+1}{(r+1)^2}&lt;/math&gt;.<br /> <br /> ==Proof 4==<br /> Proof by ishanvannadil2008:<br /> <br /> Just use jayasharmaramankumarguptareddybavarajugopal's lemma. (Thanks to tenebrine)<br /> <br /> =Application 1=<br /> ==Problem==<br /> The Wooga Looga Theorem states that the solution to this problem by franzliszt:<br /> <br /> In &lt;math&gt;\triangle ABC&lt;/math&gt; points &lt;math&gt;X,Y,Z&lt;/math&gt; are on sides &lt;math&gt;BC,CA,AB&lt;/math&gt; such that &lt;math&gt;\frac{XB}{XC}=\frac{YC}{YA}=\frac{ZA}{ZB}=\frac 71&lt;/math&gt;. Find the ratio of &lt;math&gt;[XYZ]&lt;/math&gt; to &lt;math&gt;[ABC]&lt;/math&gt;.<br /> <br /> ==Solution 1==<br /> One solution is this one by [[User:RedFireTruck|&lt;font color=&quot;#FF0000&quot;&gt;RedFireTruck&lt;/font&gt;]] ([[User talk:RedFireTruck|&lt;font color=&quot;#FF0000&quot;&gt;talk&lt;/font&gt;]]) 12:11, 1 February 2021 (EST):<br /> <br /> WLOG let &lt;math&gt;A=(0, 0)&lt;/math&gt;, &lt;math&gt;B=(1, 0)&lt;/math&gt;, &lt;math&gt;C=(x, y)&lt;/math&gt;. Then &lt;math&gt;[ABC]=\frac12|y|&lt;/math&gt; by Shoelace Theorem and &lt;math&gt;X=\left(\frac{7x+1}{8}, \frac{7y}{8}\right)&lt;/math&gt;, &lt;math&gt;Y=\left(\frac{x}{8}, \frac{y}{8}\right)&lt;/math&gt;, &lt;math&gt;Z=\left(\frac78, 0\right)&lt;/math&gt;. Then &lt;math&gt;[XYZ]=\frac12\left|\frac{43y}{64}\right|&lt;/math&gt; by Shoelace Theorem. Therefore the answer is &lt;math&gt;\boxed{\frac{43}{64}}&lt;/math&gt;.<br /> <br /> ==Solution 2==<br /> or this solution by franzliszt:<br /> <br /> We apply Barycentric Coordinates w.r.t. &lt;math&gt;\triangle ABC&lt;/math&gt;. Let &lt;math&gt;A=(1,0,0),B=(0,1,0),C=(0,0,1)&lt;/math&gt;. Then we find that &lt;math&gt;X=\left(0,\tfrac 18,\tfrac 78\right),Y=\left(\tfrac 78,0,\tfrac 18\right),Z=\left(\tfrac18,\tfrac78,0\right)&lt;/math&gt;. In the barycentric coordinate system, the area formula is &lt;math&gt;[XYZ]=\begin{vmatrix}<br /> x_{1} &amp;y_{1} &amp;z_{1} \\<br /> x_{2} &amp;y_{2} &amp;z_{2} \\ <br /> x_{3}&amp; y_{3} &amp; z_{3}<br /> \end{vmatrix}\cdot [ABC]&lt;/math&gt; where &lt;math&gt;\triangle XYZ&lt;/math&gt; is a random triangle and &lt;math&gt;\triangle ABC&lt;/math&gt; is the reference triangle. Using this, we find that &lt;cmath&gt;\frac{[XYZ]}{[ABC]}=\begin{vmatrix}<br /> 0&amp;\tfrac 18&amp;\tfrac 78\\<br /> \tfrac 78&amp;0&amp;\tfrac 18\\<br /> \tfrac18&amp;\tfrac78&amp;0<br /> \end{vmatrix}=\frac{43}{64}.&lt;/cmath&gt; &lt;math&gt;\blacksquare&lt;/math&gt;<br /> ==Solution 3==<br /> or this solution by aaja3427:<br /> <br /> According the the Wooga Looga Theorem, It is &lt;math&gt;\frac{49-7+1}{8^2}&lt;/math&gt;. This is &lt;math&gt;\boxed{\frac{43}{64}}&lt;/math&gt;<br /> <br /> ==Solution 4==<br /> <br /> or this solution by AoPS user ilovepizza2020:<br /> <br /> We use the &lt;math&gt;\mathbf{FUNDAMENTAL~THEOREM~OF~GEOGEBRA}&lt;/math&gt; to instantly get &lt;math&gt;\boxed{\frac{43}{64}}&lt;/math&gt;. (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.)<br /> <br /> ==Solution 5==<br /> or this solution by eduD_looC:<br /> <br /> This is a perfect application of the Adihaya Jayasharmaramankumarguptareddybavarajugopal's Lemma, which results in the answer being &lt;math&gt;\boxed{\frac{43}{64}}&lt;/math&gt;. A very beautiful application, which leaves graders and readers speechless. Great for math contests with proofs.<br /> <br /> ==Solution 6==<br /> or this solution by CoolJupiter:<br /> <br /> Wow. All of your solutions are slow, compared to my sol:<br /> <br /> By math, we have &lt;math&gt;\boxed{\frac{43}{64}}&lt;/math&gt;.<br /> <br /> ~CoolJupiter<br /> ^<br /> |<br /> EVERYONE USE THIS SOLUTION IT'S BRILLIANT <br /> ~bsu1<br /> Yes, very BRILLIANT!<br /> ~ TheAoPSLebron<br /> <br /> ==The Best Solution==<br /> <br /> By the &lt;math&gt;1+1=\text{BREAD}&lt;/math&gt; principle, we get &lt;math&gt;\boxed{\frac{43}{64}}&lt;/math&gt;. Definitely the best method. When asked, please say that OlympusHero taught you this method. Because he did.<br /> <br /> ==Easiest Solution==<br /> <br /> The answer is clearly &lt;math&gt;\boxed{\frac{43}{64}}&lt;/math&gt;. We leave the proof and intermediate steps to the reader as an exercise.<br /> <br /> =Application 2=<br /> ==Problem==<br /> The Wooga Looga Theorem states that the solution to this problem by Matholic:<br /> <br /> The figure below shows a triangle ABC whose area is &lt;math&gt;72 \text{cm}^2&lt;/math&gt;. If &lt;math&gt;\dfrac{AD}{DB}=\dfrac{BE}{EC}=\dfrac{CF}{FA}=\dfrac{1}{5}&lt;/math&gt;, find &lt;math&gt;[DEF].&lt;/math&gt;<br /> <br /> ~LaTeX-ifyed by RP3.1415<br /> <br /> ==Solution 1==<br /> is this solution by franzliszt:<br /> <br /> We apply Barycentric Coordinates w.r.t. &lt;math&gt;\triangle ABC&lt;/math&gt;. Let &lt;math&gt;A=(1,0,0),B=(0,1,0),C=(0,0,1)&lt;/math&gt;. Then we find that &lt;math&gt;D=(\tfrac 56,\tfrac 16,0),E=(0,\tfrac 56,\tfrac 16),F=(\tfrac16,0,\tfrac56)&lt;/math&gt;. In the barycentric coordinate system, the area formula is &lt;math&gt;[XYZ]=\begin{vmatrix}<br /> x_{1} &amp;y_{1} &amp;z_{1} \\ <br /> x_{2} &amp;y_{2} &amp;z_{2} \\ <br /> x_{3}&amp; y_{3} &amp; z_{3}<br /> \end{vmatrix}\cdot [ABC]&lt;/math&gt; where &lt;math&gt;\triangle XYZ&lt;/math&gt; is a random triangle and &lt;math&gt;\triangle ABC&lt;/math&gt; is the reference triangle. Using this, we find that&lt;cmath&gt;\frac{[DEF]}{}=\begin{vmatrix}<br /> \tfrac 56&amp;\tfrac 16&amp;0\\<br /> 0&amp;\tfrac 56&amp;\tfrac 16\\<br /> \tfrac16&amp;0&amp;\tfrac56<br /> \end{vmatrix}=\frac{7}{12}&lt;/cmath&gt; so &lt;math&gt;[DEF]=42&lt;/math&gt;. &lt;math&gt;\blacksquare&lt;/math&gt;<br /> ==Solution 2==<br /> or this solution by [[User:RedFireTruck|&lt;font color=&quot;#FF0000&quot;&gt;RedFireTruck&lt;/font&gt;]] ([[User talk:RedFireTruck|&lt;font color=&quot;#FF0000&quot;&gt;talk&lt;/font&gt;]]) 12:11, 1 February 2021 (EST):<br /> <br /> By the Wooga Looga Theorem, &lt;math&gt;\frac{[DEF]}{[ABC]}=\frac{5^2-5+1}{(5+1)^2}=\frac{21}{36}=\frac{7}{12}&lt;/math&gt;. We are given that &lt;math&gt;[ABC]=72&lt;/math&gt; so &lt;math&gt;[DEF]=\frac{7}{12}\cdot72=\boxed{42}&lt;/math&gt;<br /> <br /> =Application 3=<br /> ==Problem==<br /> The Wooga Looga Theorem states that the solution to this problem by RedFireTruck:<br /> <br /> Find the ratio &lt;math&gt;\frac{[GHI]}{[ABC]}&lt;/math&gt; if &lt;math&gt;\frac{AD}{DB}=\frac{BE}{EC}=\frac{CF}{FA}=\frac12&lt;/math&gt; and &lt;math&gt;\frac{DG}{GE}=\frac{EH}{HF}=\frac{FI}{ID}=1&lt;/math&gt; in the diagram below.&lt;asy&gt;<br /> draw((0, 0)--(6, 0)--(4, 3)--cycle);<br /> draw((2, 0)--(16/3, 1)--(8/3, 2)--cycle);<br /> draw((11/3, 1/2)--(4, 3/2)--(7/3, 1)--cycle);<br /> label(&quot;$A$&quot;, (0, 0), SW);<br /> label(&quot;$B$&quot;, (6, 0), SE);<br /> label(&quot;$C$&quot;, (4, 3), N);<br /> label(&quot;$D$&quot;, (2, 0), S);<br /> label(&quot;$E$&quot;, (16/3, 1), NE);<br /> label(&quot;$F$&quot;, (8/3, 2), NW);<br /> label(&quot;$G$&quot;, (11/3, 1/2), SE);<br /> label(&quot;$H$&quot;, (4, 3/2), NE);<br /> label(&quot;$I$&quot;, (7/3, 1), W);<br /> &lt;/asy&gt;<br /> ==Solution 1==<br /> is this solution by franzliszt:<br /> <br /> By the Wooga Looga Theorem, &lt;math&gt;\frac{[DEF]}{[ABC]}=\frac{2^2-2+1}{(1+2)^2}=\frac 13&lt;/math&gt;. Notice that &lt;math&gt;\triangle GHI&lt;/math&gt; is the medial triangle of '''Wooga Looga Triangle ''' of &lt;math&gt;\triangle ABC&lt;/math&gt;. So &lt;math&gt;\frac{[GHI]}{[DEF]}=\frac 14&lt;/math&gt; and &lt;math&gt;\frac{[GHI]}{[ABD]}=\frac{[DEF]}{[ABC]}\cdot\frac{[GHI]}{[DEF]}=\frac 13 \cdot \frac 14 = \frac {1}{12}&lt;/math&gt; by Chain Rule ideas.<br /> <br /> ==Solution 2==<br /> or this solution by franzliszt:<br /> <br /> Apply Barycentrics w.r.t. &lt;math&gt;\triangle ABC&lt;/math&gt; so that &lt;math&gt;A=(1,0,0),B=(0,1,0),C=(0,0,1)&lt;/math&gt;. Then &lt;math&gt;D=(\tfrac 23,\tfrac 13,0),E=(0,\tfrac 23,\tfrac 13),F=(\tfrac 13,0,\tfrac 23)&lt;/math&gt;. And &lt;math&gt;G=(\tfrac 13,\tfrac 12,\tfrac 16),H=(\tfrac 16,\tfrac 13,\tfrac 12),I=(\tfrac 12,\tfrac 16,\tfrac 13)&lt;/math&gt;.<br /> <br /> In the barycentric coordinate system, the area formula is &lt;math&gt;[XYZ]=\begin{vmatrix} x_{1} &amp;y_{1} &amp;z_{1} \\ x_{2} &amp;y_{2} &amp;z_{2} \\ x_{3}&amp; y_{3} &amp; z_{3} \end{vmatrix}\cdot [ABC]&lt;/math&gt; where &lt;math&gt;\triangle XYZ&lt;/math&gt; is a random triangle and &lt;math&gt;\triangle ABC&lt;/math&gt; is the reference triangle. Using this, we find that&lt;cmath&gt;\frac{[GHI]}{[ABC]}=\begin{vmatrix} \tfrac 13&amp;\tfrac 12&amp;\tfrac 16\\ \tfrac 16&amp;\tfrac 13&amp;\tfrac 12\\ \tfrac 12&amp;\tfrac 16&amp;\tfrac 13 \end{vmatrix}=\frac{1}{12}.&lt;/cmath&gt;<br /> <br /> =Application 4=<br /> ==Problem==<br /> <br /> Let &lt;math&gt;ABC&lt;/math&gt; be a triangle and &lt;math&gt;D,E,F&lt;/math&gt; be points on sides &lt;math&gt;BC,AC,&lt;/math&gt; and &lt;math&gt;AB&lt;/math&gt; respectively. We have that &lt;math&gt;\frac{BD}{DC} = 3&lt;/math&gt; and similar for the other sides. If the area of triangle &lt;math&gt;ABC&lt;/math&gt; is &lt;math&gt;16&lt;/math&gt;, then what is the area of triangle &lt;math&gt;DEF&lt;/math&gt;? (By ilovepizza2020)<br /> <br /> ==Solution 1==<br /> <br /> By Franzliszt<br /> <br /> By Wooga Looga, &lt;math&gt;\frac{[DEF]}{16} = \frac{3^2-3+1}{(3+1)^2}=\frac{7}{16}&lt;/math&gt; so the answer is &lt;math&gt;\boxed7&lt;/math&gt;.<br /> <br /> ==Solution 2==<br /> <br /> By franzliszt<br /> <br /> Apply Barycentrics w.r.t. &lt;math&gt;\triangle ABC&lt;/math&gt;. Then &lt;math&gt;A=(1,0,0),B=(0,1,0),C=(0,0,1)&lt;/math&gt;. We can also find that &lt;math&gt;D=(0,\tfrac 14,\tfrac 34),E=(\tfrac 34,0,\tfrac 14),F=(\tfrac 14,\tfrac 34,0)&lt;/math&gt;. In the barycentric coordinate system, the area formula is &lt;math&gt;[XYZ]=\begin{vmatrix} x_{1} &amp;y_{1} &amp;z_{1} \\ x_{2} &amp;y_{2} &amp;z_{2} \\ x_{3}&amp; y_{3} &amp; z_{3} \end{vmatrix}\cdot [ABC]&lt;/math&gt; where &lt;math&gt;\triangle XYZ&lt;/math&gt; is a random triangle and &lt;math&gt;\triangle ABC&lt;/math&gt; is the reference triangle. Using this, we find that&lt;cmath&gt;\frac{[DEF]}{[ABC]}=\begin{vmatrix} 0&amp;\tfrac 14&amp;\tfrac 34\\ \tfrac 34&amp;0&amp;\tfrac 14\\ \tfrac 14&amp;\tfrac 34&amp;0 \end{vmatrix}=\frac{7}{16}.&lt;/cmath&gt;So the answer is &lt;math&gt;\boxed{7}&lt;/math&gt;.<br /> <br /> ==Solution 3==<br /> <br /> A long story short, the answer must be &lt;math&gt;\boxed{7}&lt;/math&gt; by the inverse of the Inverse Wooga Looga Theorem<br /> <br /> =Testimonials=<br /> <br /> Pogpr0 = wooga looga - Ladka13<br /> The Wooga Looga Theorem can be used to prove many problems and should be a part of any geometry textbook.<br /> ~ilp2020<br /> <br /> Thanks for rediscovering our theorem RedFireTruck - Foogle and Hoogle of [https://www.youtube.com/channel/UC50E9TuLIMWbOPUX45xZPaQ The Ooga Booga Tribe of The Caveman Society]<br /> <br /> Franzlist is wooga looga howsopro - volkie boy<br /> <br /> this is in fact a pretty sensible theorem. Nothing to be so excited about, though. ~DofL<br /> <br /> The Wooga Looga Theorem is EPIC POGGERS WHOLESOME 100 KEANU CHUNGUS AMAZING SKILL THEOREM!!!!!1!!!111111 -centslordm<br /> <br /> The Wooga Looga Theorem is amazing and can be applied to so many problems and should be taught in every school. - [[User:RedFireTruck|&lt;font color=&quot;#FF0000&quot;&gt;RedFireTruck&lt;/font&gt;]] ([[User talk:RedFireTruck|&lt;font color=&quot;#FF0000&quot;&gt;talk&lt;/font&gt;]]) 11:00, 1 February 2021 (EST)<br /> <br /> The Wooga Looga Theorem is the best. -aaja3427<br /> <br /> The Wooga Looga Theorem is needed for everything and it is great-hi..<br /> <br /> 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 &quot;about&quot;. 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<br /> <br /> This theorem has helped me with school and I am no longer failing my math class. -mchang<br /> <br /> 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<br /> <br /> &quot;I can't believe AoPS books don't have this amazing theorem. If you need help with math, you can depend on caveman.&quot; ~CoolJupiter<br /> <br /> 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)<br /> <br /> Too powerful... ~franzliszt<br /> <br /> The Wooga Looga Theorem is so pro ~ ac142931<br /> <br /> It is so epic and awesome that it will blow the minds of people if they saw this ~ ac142931(2nd testimonial by me)<br /> <br /> This theorem changed my life... ~ samrocksnature<br /> <br /> Math competitions need to ban the use of the Wooga Looga Theorem, it's just too good. ~ jasperE3<br /> <br /> It actually can be. I never thought I'd say this, but the Wooga Looga theorem is a legit theorem. ~ jasperE3<br /> <br /> This is franzliszt and I endorse this theorem. ~franzliszt<br /> <br /> This theorem is too OP. ~bestzack66<br /> <br /> This is amazing! However much it looks like a joke, it is a legitimate - and powerful - theorem. -Supernova283<br /> <br /> 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<br /> <br /> The Wooga Looga cavemen were way ahead of their time. Good job (dead) guys! -HIA2020<br /> <br /> It's like the Ooga Booga Theorem (also OP), but better!!! - BobDBuilder321<br /> <br /> 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<br /> <br /> I actually thought this was a joke theorem until I read this page - HumanCalculator9<br /> <br /> I endorse the Wooga Looga theorem for its utter usefulness and seriousness. -HamstPan38825<br /> <br /> This is &lt;i&gt;almost&lt;/i&gt; as OP as the Adihaya Jayasharmaramankumarguptareddybavarajugopal Lemma. Needs to be nerfed. -CoolCarsonTheRun<br /> <br /> &lt;s&gt;I ReAlLy don't get it - Senguamar&lt;/s&gt; HOW DARE YOU!!!!<br /> <br /> The Wooga Looga Theorem is the base of all geometry. It is so OP that even I don't understand how to use it.<br /> <br /> You know what, this is jayasharmaramankumarguptareddybavarajugopal's lemma - Ishan<br /> <br /> If only I knew this on some contests that I had done previously... - JacobJB<br /> <br /> The Wooga Looga Theorem is so pr0 that it needs to be nerfed. - rocketsri<br /> <br /> &quot;The Wooga Looga Theorem should be used in contests and should be part of geometry books.&quot; ~ [[User:Aops-g5-gethsemanea2|Aops-g5-gethsemanea2]] ([[User talk:Aops-g5-gethsemanea2|talk]]) 21:56, 21 December 2020 (EST)<br /> <br /> The Wooga Looga Theorem is so OP BRUH<br /> <br /> 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<br /> <br /> I have no idea what is going on here - awesomeguy856<br /> <br /> fuzimiao2013 waz hear<br /> <br /> this theorem is bad<br /> <br /> poggers theorem - awesomeming327<br /> <br /> The Wooga Looga theorem is very OP and not to be frowned upon - Yelly314<br /> <br /> 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<br /> <br /> Wooga Looga Theorem is TRASH.<br /> <br /> 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.<br /> <br /> Now I know how @Louis_Vuitton got so much smarter than me! :rotfl:</div> Nodens https://artofproblemsolving.com/wiki/index.php?title=User:Nodens&diff=154109 User:Nodens 2021-05-22T22:13:50Z <p>Nodens: </p> <hr /> <div>This is my personal page, i created it because of [redacted] and [redacted]<br /> <br /> I live in [data expunged] and like [redacted] and [wsmdifk34i5u84392oieudiokmdjweiol~data error!]<br /> <br /> I have not done [redacted] competition and generally don't do [data expunged] unless i have to.....<br /> <br /> WAIT how did you find this? Are you stalking me? o.O<br /> <br /> Ps: [redacted], [redacted], [data expunged]<br /> <br /> P.P.s: We're no strangers to love<br /> You know the rules and so do i<br /> A full commitment's what i'm looking at<br /> You wouldn't get this from any other guy<br /> I just wanna tell you how i'm feelin<br /> Gotta make you understand<br /> <br /> jk, i accept friend requests. :)</div> Nodens https://artofproblemsolving.com/wiki/index.php?title=User:Nodens&diff=154108 User:Nodens 2021-05-22T22:13:08Z <p>Nodens: </p> <hr /> <div>This is my personal page, i created it because of [redacted] and [redacted]<br /> <br /> I live in [data expunged] and like [redacted] and [wsmdifk34i5u84392oieudiokmdjweiol~data error!]<br /> <br /> I have not done [redacted] competition and generally don't do [data expunged] unless i have to.....<br /> <br /> WAIT how did you find this? Are you stalking me? o.0<br /> <br /> Ps: [redacted], [redacted], [data expunged]<br /> <br /> P.P.s: We're no strangers to love<br /> You know the rules and so do i<br /> A full commitment's what i'm looking at<br /> You won't get this from any other guy<br /> I just wanna tell you how i'm feelin<br /> Gotta make you understand<br /> <br /> jk, i accept friend requests. :)</div> Nodens https://artofproblemsolving.com/wiki/index.php?title=User:Nodens&diff=154107 User:Nodens 2021-05-22T22:12:57Z <p>Nodens: </p> <hr /> <div>This is my personal page, i created it because of [redacted] and [redacted]<br /> <br /> I live in [data expunged] and like [redacted] and [wsmdifk34i5u84392oieudiokmdjweiol~data error!]<br /> <br /> I have not done [redacted] competition and generally don't do [data expunged] unless i have to.....<br /> <br /> WAIT how did you find this? Are you stalking me? o.O<br /> <br /> Ps: [redacted], [redacted], [data expunged]<br /> <br /> P.P.s: We're no strangers to love<br /> You know the rules and so do i<br /> A full commitment's what i'm looking at<br /> You won't get this from any other guy<br /> I just wanna tell you how i'm feelin<br /> Gotta make you understand<br /> <br /> jk, i accept friend requests. :)</div> Nodens https://artofproblemsolving.com/wiki/index.php?title=User:OlympusHero&diff=153344 User:OlympusHero 2021-05-07T22:54:42Z <p>Nodens: /* User Count */</p> <hr /> <div>OlympusHero's Page:<br /> &lt;br&gt;<br /> __NOTOC__&lt;div style=&quot;border:2px solid black; -webkit-border-radius: 10px; background:#dddddd&quot;&gt;<br /> ==&lt;font color=&quot;black&quot; style=&quot;font-family: ITC Avant Garde Gothic Std, Verdana&quot;&gt;&lt;div style=&quot;margin-left:10px&quot;&gt;User Count&lt;/div&gt;&lt;/font&gt;==<br /> &lt;div style=&quot;margin-left: 10px; margin-bottom:10px&quot;&gt;&lt;font color=&quot;black&quot;&gt;If this is your first time visiting this page, edit it by incrementing the user count below by one.&lt;/font&gt;&lt;/div&gt;<br /> &lt;center&gt;&lt;font size=&quot;107px&quot;&gt; 82 users have visted this!<br /> &lt;/font&gt;&lt;/center&gt;<br /> &lt;/div&gt;<br /> &lt;div style=&quot;border:2px solid black; background:#cccccc;-webkit-border-radius: 10px; align:center&quot;&gt;<br /> <br /> ==&lt;font color=&quot;black&quot; style=&quot;font-family: ITC Avant Garde Gothic Std, Verdana&quot;&gt;&lt;div style=&quot;margin-left:10px&quot;&gt;About Me&lt;/div&gt;&lt;/font&gt;==<br /> &lt;div style=&quot;margin-left: 10px; margin-bottom:10px&quot;&gt;&lt;font color=&quot;black&quot;&gt;OlympusHero is currently borderline AIME.&lt;br&gt;<br /> <br /> OlympusHero is 10 years old.&lt;br&gt;<br /> <br /> OlympusHero scored 47/46 (got an extra point for finishing the test under 10 minutes) when mocking the 2019 MATHCOUNTS State test, and got silver on the 2020 online MATHCOUNTS State.&lt;br&gt;<br /> <br /> OlympusHero is a pro at math and chess<br /> <br /> OlympusHero for brown MOP 2021<br /> <br /> OlympusHero was #1 at the 2017 Chess World Cadets U8 (He was 7 years old)<br /> <br /> OlympusHero is pro at joking believe it.<br /> <br /> OlympusHero is much better than Rusczyk at math yes or no, no one knows. (he's not)<br /> <br /> -----------<br /> <br /> OlympusHero is very pro and is much better than bestzack66 in MC trainer, Alcumus, and FTW. ~bestzack66<br /> &lt;/font&gt;&lt;/div&gt;<br /> &lt;/div&gt;<br /> &lt;div style=&quot;border:2px solid black; background:#bbbbbb;-webkit-border-radius: 10px; align:center&quot;&gt;<br /> <br /> ==&lt;font color=&quot;black&quot; style=&quot;font-family: ITC Avant Garde Gothic Std, Verdana&quot;&gt;&lt;div style=&quot;margin-left:10px&quot;&gt;Goals&lt;/div&gt;&lt;/font&gt;==<br /> &lt;div style=&quot;margin-left: 10px; margin-right: 10px; margin-bottom:10px&quot;&gt;A User Count of 300<br /> <br /> Make AIME 2021 (Currently borderline)<br /> <br /> Get 5 or more on AIME I 2021 (Mocked a 3 on the 2020 AIME I a few months ago)<br /> <br /> - some months later; guess I have improved (got from 3--&gt;6 for 2020 AIME I) sadly sillies killed me crap<br /> <br /> Get to 40 or more on the MATHCOUNTS Trainer Nationals Level (25/40)<br /> <br /> Get a perfect score and First Place in IMO 2021.<br /> &lt;/div&gt;<br /> &lt;/div&gt;</div> Nodens https://artofproblemsolving.com/wiki/index.php?title=User:Nodens&diff=152906 User:Nodens 2021-04-28T18:27:40Z <p>Nodens: </p> <hr /> <div>This is my personal page, i created it because of [redacted] and [redacted]<br /> <br /> I live in [data expunged] and like [redacted] and [wsmdifk34i5u84392oieudiokmdjweiol~data error!]<br /> <br /> I have not done [redacted] competition and generally don't do [data expunged] unless i have to.....<br /> <br /> WAIT how did you find this? Are you stalking me? o_O<br /> <br /> Ps: [redacted], [redacted], [data expunged]<br /> <br /> P.P.s: We're no strangers to love<br /> You know the rules and so do i<br /> A full commitment's what i'm looking at<br /> You won't get this from any other guy<br /> I just wanna tell you how i'm feelin<br /> Gotta make you understand<br /> <br /> jk, i accept friend requests. :)</div> Nodens https://artofproblemsolving.com/wiki/index.php?title=User:Nodens&diff=152263 User:Nodens 2021-04-21T19:50:10Z <p>Nodens: </p> <hr /> <div>This is my personal page, i created it because of [redacted] and [redacted]<br /> <br /> I live in [data expunged] and like [redacted] and [wsmdifk34i5u84392oieudiokmdjweiol~data error!]<br /> <br /> I have not done [redacted] competition and generally don't do [data expunged] unless i have to.....<br /> <br /> WAIT how did you find this? Are you stalking me? o_O<br /> <br /> Ps: [redacted], [redacted], [data expunged]<br /> <br /> <br /> jk, i accept friend requests. :)</div> Nodens https://artofproblemsolving.com/wiki/index.php?title=User:Centslordm&diff=152049 User:Centslordm 2021-04-20T22:20:04Z <p>Nodens: </p> <hr /> <div>[https://avatar.artofproblemsolving.com/avatar_430228.png avatar]<br /> <br /> centslordm is postbanned for spam! he is a spammer. pretty bad at it. asdlkfj<br /> <br /> centslordm has a spam blog that was originally meant for actual quality until he added a billion contributors. everything devolved from there. it now has ~16k visits (oops, now 18k! (oops, now 24k! (oops, now 25k!(oops, now 35k!!)))). lol. it gets like 300 visits a day for some reason that is unknown to centslordm. <br /> <br /> centslordm has 4k upvotes. he is one of the elite few who have 4k+ upvotes. <br /> <br /> centslordm has 10k upvotes. he is one of the elite few who have 10k+ upvotes B)<br /> <br /> centslordm postfarmed to 1000 posts. he now regrets that. if he didn't postfarm he would probably have around 900 posts total right now. welp<br /> <br /> centslordm AIME qualled in 6th grade, with a score of 105. he cheesed by doin 15 problems correctly while leaving 10. he hopes to change that this year. LOL<br /> <br /> centslordm has over 3000 friends on AOPS<br /> <br /> centslordm Is known by pretty much everyone on AOPS<br /> <br /> centslordm wastes time a lot. the ultimate procrastinator.<br /> <br /> centslordm has over 2000 posts<br /> <br /> centslordm is the ULTIMATE simp. <br /> Visit his second blog for CSS: https://artofproblemsolving.com/community/c1621357h2356796_test_post<br /> <br /> centslordm has his own forum, EMNC, a forum where everyone centsfarms. it is currently in the top 20 forums. <br /> <br /> centslordm is the founder of The Brawl Stars Forum, Everything Makes No Cents, The Doki Doki Literature Club Forum, and The Friending Forum!<br /> <br /> centslordm speedran minecraft in 2nd grade. it took him 3 days, and he played for ~1 hr and 15 min a day. So, he beat minecraft in 3 hrs and 45 minutes then.<br /> <br /> centslordm came up with his username in 2nd grade, too. &quot;CentsLord&quot; was a unique and cool name, and it still is. However, the letter &quot;m&quot; was added unto the username while making the AoPS account as Monika had tampered with the interface and insisted on having her initial in centslordm's username. (Not really, but it's a widely-known legend. Don't get jealous, simps!)<br /> <br /> centslordm is actually Dream in disguise.<br /> <br /> &lt;p style=&quot;color:red;&quot;&gt;Who is Dream? I never &lt;span style=&quot;font-style: italic&quot;&gt;dreamed&lt;/span&gt; of him...get it? Because whoever that is, is named Dream. :P &lt;/p&gt;<br /> <br /> Cents does not know who samuel1234 is. This is because samuel1234 is the most glerpy user on AoPS. samuel1234 always says &quot;kawepp&quot; for some reason, which censtlordm does not understand. If he did understand, he could rival GMAAS.<br /> <br /> from cents- Samuel, I DO know who you are XDDD you are NOT the most glerpy user on AoPS. Kawepp is life!<br /> <br /> from samuel1234 - How can you prove you know me? (Scared noises)<br /> <br /> because you are cool B)<br /> <br /> centslordm wants to get on the HoF soon, but doesn't really try lol. he has a decent-ish alcumus profile, with a 99 something being his highest rating in C&amp;P. centslordm hates geometry.<br /> <br /> centslordm just reset his alcumus data<br /> <br /> centslordm also does not know the user Nodens, despite them being active on EMNC, his blog, and rickrolling him several times.</div> Nodens https://artofproblemsolving.com/wiki/index.php?title=User:Piphi&diff=152048 User:Piphi 2021-04-20T22:13:13Z <p>Nodens: </p> <hr /> <div>{{User:Piphi/Template:Header}}<br /> &lt;br&gt;<br /> __NOTOC__&lt;div style=&quot;border:2px solid black; -webkit-border-radius: 10px; background:#F0F2F3&quot;&gt;<br /> ==&lt;font color=&quot;black&quot; style=&quot;font-family: ITC Avant Garde Gothic Std, Verdana&quot;&gt;&lt;div style=&quot;margin-left:10px&quot;&gt;User Count&lt;/div&gt;&lt;/font&gt;==<br /> &lt;div style=&quot;margin-left: 10px; margin-bottom:10px&quot;&gt;&lt;font color=&quot;black&quot;&gt;If this is your first time visiting this page, edit it by incrementing the user count below by one.&lt;/font&gt;&lt;/div&gt;<br /> &lt;center&gt;&lt;font size=&quot;100px&quot;&gt;486 views&lt;/font&gt;&lt;/center&gt;<br /> &lt;/div&gt;<br /> &lt;div style=&quot;border:2px solid black; background:#919293;-webkit-border-radius: 10px; align:center&quot;&gt;<br /> &lt;!-- &lt;div&gt;Q: How to annoy a web developer?&lt;/span&gt; :P --&gt;<br /> <br /> ==&lt;font color=&quot;black&quot; style=&quot;font-family: ITC Avant Garde Gothic Std, Verdana&quot;&gt;&lt;div style=&quot;margin-left:10px&quot;&gt;About Me&lt;/div&gt;&lt;/font&gt;==<br /> &lt;div style=&quot;margin-left: 10px; margin-bottom:10px&quot;&gt;&lt;font color=&quot;black&quot;&gt;&lt;font color=&quot;black&quot;&gt;<br /> Piphi created [[User:Piphi/Games|AoPS Wiki Games by Piphi]].&lt;br&gt;<br /> <br /> Piphi has been very close to winning multiple [[Greed Control]] games, piphi placed 5th in game #18 and 2nd in game #19. Thanks to piphi, Greed Control games have started to be kept track of. Piphi made a spreadsheet that has all of Greed Control history [https://artofproblemsolving.com/community/c19451h2126208p15569802 here].&lt;br&gt;<br /> <br /> Piphi has a side-project that is making the Wiki's [[Main Page]] look better, you can check that out [[User:Piphi/AoPS Wiki|here]].&lt;br&gt;<br /> <br /> Piphi is a proud member of [https://artofproblemsolving.com/community/c562043 The Interuniversal GMAAS Society].<br /> <br /> Piphi is part of the [https://artofproblemsolving.com/community/c1124279 Asymptote Competition] staff.<br /> &lt;/font&gt;&lt;/div&gt;<br /> &lt;/div&gt;<br /> &lt;div style=&quot;border:2px solid black; background:#333333;-webkit-border-radius: 10px; align:center&quot;&gt;<br /> <br /> ==&lt;font color=&quot;#f0f2f3&quot; style=&quot;font-family: ITC Avant Garde Gothic Std, Verdana&quot;&gt;&lt;div style=&quot;margin-left:10px&quot;&gt;Goals&lt;/div&gt;&lt;/font&gt;==<br /> &lt;div style=&quot;margin-left: 10px; margin-right: 10px; margin-bottom:10px&quot;&gt;&lt;font color=&quot;#f0f2f3&quot;&gt;<br /> You can check out more goals/statistics [[User:Piphi/Statistics|here]].<br /> <br /> A User Count of 500<br /> {{User:Piphi/Template:Progress_Bar|96.4|width=100%}}<br /> <br /> 200 subpages of [[User:Piphi]]<br /> {{User:Piphi/Template:Progress_Bar|99|width=100%}}<br /> <br /> 200 signups for [[User:Piphi/Games|AoPS Wiki Games by Piphi]]<br /> {{User:Piphi/Template:Progress_Bar|54|width=100%}}<br /> <br /> Make 10,000 edits<br /> {{User:Piphi/Template:Progress_Bar|22.81|width=100%}}&lt;/font&gt;&lt;/div&gt;<br /> &lt;/div&gt;</div> Nodens https://artofproblemsolving.com/wiki/index.php?title=User:Nodens&diff=151981 User:Nodens 2021-04-19T23:12:07Z <p>Nodens: </p> <hr /> <div>This is my personal page, i created it because of [redacted] and [redacted]<br /> <br /> I live in [data expunged] and like [redacted] and [wsmdifk34i5u84392oieudiokmdjweiol~data error!]<br /> <br /> I have not done [redacted] competition and generally don't do [data expunged] unless i have to.....<br /> <br /> WAIT how did you find this? Are you stalking me? o_O<br /> <br /> <br /> jk, i accept friend requests. :)</div> Nodens https://artofproblemsolving.com/wiki/index.php?title=User:Nodens&diff=151980 User:Nodens 2021-04-19T23:11:51Z <p>Nodens: </p> <hr /> <div>[category:Nodens]<br /> <br /> This is my personal page, i created it because of [redacted] and [redacted]<br /> <br /> I live in [data expunged] and like [redacted] and [wsmdifk34i5u84392oieudiokmdjweiol~data error!]<br /> <br /> I have not done [redacted] competition and generally don't do [data expunged] unless i have to.....<br /> <br /> WAIT how did you find this? Are you stalking me? o_O<br /> <br /> <br /> jk, i accept friend requests. :)</div> Nodens https://artofproblemsolving.com/wiki/index.php?title=User:Nodens&diff=151731 User:Nodens 2021-04-17T00:44:42Z <p>Nodens: </p> <hr /> <div>This is my personal page, i created it because of [redacted] and [redacted]<br /> <br /> I live in [data expunged] and like [redacted] and [wsmdifk34i5u84392oieudiokmdjweiol~data error!]<br /> <br /> I have not done [redacted] competition and generally don't do [data expunged] unless i have to.....<br /> <br /> WAIT how did you find this? Are you stalking me? o_O<br /> <br /> <br /> jk, i accept friend requests. :)</div> Nodens https://artofproblemsolving.com/wiki/index.php?title=User:Nodens&diff=151730 User:Nodens 2021-04-17T00:44:05Z <p>Nodens: </p> <hr /> <div>This is my personal page, i created it because of [redacted] and [redacted]<br /> <br /> I live in [data expunged] and like [redacted] and [wsmdifk34i5u84392oieudiokmdjweiol~data error!]<br /> <br /> I have not done [redacted] competition and generally don't do [data expunged] unless i have to.....<br /> <br /> [b][size=250]WAIT how did you find this? Are you stalking me? o_O[/size][/b]<br /> <br /> <br /> [size=25]jk, i accept friend requests.[/size]</div> Nodens https://artofproblemsolving.com/wiki/index.php?title=User:Nodens&diff=151729 User:Nodens 2021-04-17T00:41:08Z <p>Nodens: </p> <hr /> <div>This is my personal page, i created it because of [redacted] and [redacted]<br /> <br /> I live in [data expunged] and like [redacted] and [wsmdifk34i5u84392oieudiokmdjweiol~data error!]</div> Nodens https://artofproblemsolving.com/wiki/index.php?title=User:Nodens&diff=151604 User:Nodens 2021-04-15T22:06:09Z <p>Nodens: Nil</p> <hr /> <div>This is my personal page, i created it because of [redacted] and [redacted]</div> Nodens