Difference between revisions of "The Devil's Triangle"
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This is Routh's theorem isn't it~ Ilovepizza2020 | 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 | Wow this generalization of my theorem is amazing. good job. - Foogle and Hoogle, Members of the Ooga Booga Tribe of The Caveman Society |
Revision as of 19:29, 6 November 2020
Definition
==The Devil's Triangle/Routh's Theorem
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
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