# The Devil's Triangle

## Contents

# Definition

## 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)

# 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://artofproblemsolving.com/wiki/index.php/Wooga_Looga_Theorem

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)