Difference between revisions of "The Devil's Triangle"
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Wow this generalization of my theorem is amazing. good job. - Foogle and Hoogle, Members of the Ooga Booga Tribe of The Caveman Society | Wow this generalization of my theorem is amazing. good job. - Foogle and Hoogle, Members of the Ooga Booga Tribe of The Caveman Society | ||
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+ | trivial by <math>\frac{1}{2}ab\sin(C)</math> but ok ~ bissue |
Revision as of 18:22, 29 December 2020
Contents
[hide]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