The Devil's Triangle
Contents
[hide]Definition
For any triangle , let and be points on and respectively. Devil's Triangle Theorem, also known has Routh Theorem, states that if and , then , or (Shoutout to @Gogobao for pointing this out)
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 , or .
(Another shoutout to @Gogobao for the simplification, you are very helpful!!) ~@CoolJupiter
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://artofproblemsolving.com/wiki/index.php/Wooga_Looga_Theorem
Essentially, Wooga Looga is a special case of this, specifically when .
Testimonials
The Ooga Booga Tribe would be proud of you. Amazing theorem - RedFireTruck
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