The Devil's Triangle
Contents
[hide]Definition
Generalized Wooga Looga Theorem (The Devil's Triangle)
For any triangle , let and be points on and respectively. The Generalized Wooga Looga Theorem (Gwoologth) or the Devil's Triangle Theorem states that if and , then .
(*Simplification found by @Gogobao)
Proofs
Proof 1
Proof by CoolJupiter:
We have the following ratios: .
Now notice that .
We attempt to find the area of each of the smaller triangles.
Notice that using the ratios derived earlier.
Similarly, and .
Thus, .
Finally, we have .
~@CoolJupiter
Proof 2
Proof by math_comb01 Apply Barycentrics . Then . also
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 .
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 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