# The Devil's Triangle

# Definition

For any triangle , let and be points on and respectively. Devil's Triangle Theorem states that if and , then .

# 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 RedFireTruck:

WLOG let B=(1, 0)C=(x, y)xy\in\mathbb{R}$

# 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