The Devil's Triangle
Generalized Wooga Looga Theorem (The Devil's Triangle)
For any triangle , let and be points on and respectively. The Generalizwed Wooga Looga Theorem or the Devil's Triangle Theorem states that if and , then .
(*Simplification found by @Gogobao)
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 .
Finally, we have .
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 ===
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://artofproblemsolving.com/wiki/index.php/Wooga_Looga_Theorem
Essentially, Wooga Looga is a special case of this, specifically when .
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