Difference between revisions of "The Devil's Triangle"
(Generalized Wooga Looga Theorem (The Devil's Triangle)) |
Math comb01 (talk | contribs) (→Proofs) |
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~@CoolJupiter | ~@CoolJupiter | ||
+ | ==Proof 2== | ||
+ | Proof by math_comb01 | ||
+ | Apply Barycentrics w.r.t. <math>\triangle ABC</math>. Then <math>A=(1,0,0),B=(0,1,0),C=(0,0,1)</math>. We can also find that <math>D=\left(0,\tfrac {1}{r+1},\tfrac {r}{r+1}\right),E=\left(\tfrac {s}{s+1},0,\tfrac {1}{s+1}\right),F=\left(\tfrac {1}{t+1},\tfrac {t}{t+1},0\right)</math> | ||
+ | In the barycentric coordinate system, the area formula is <math>[XYZ]= | ||
+ | ~@Math_comb01 | ||
=Other Remarks= | =Other Remarks= |
Revision as of 02:23, 18 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 w.r.t. . Then . We can also find that
In the barycentric coordinate system, the area formula is where is a random triangle and is the reference triangle. Using this, we find that +\frac{rst}{([s+1][r+1][t+1]}.\]=\frac{rst+1}{([s+1][r+1][t+1]}.\]
~@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