Difference between revisions of "The Devil's Triangle"
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=Definition= | =Definition= | ||
==Generalized Wooga Looga Theorem (The Devil's Triangle)== | ==Generalized Wooga Looga Theorem (The Devil's Triangle)== | ||
− | For any triangle <math>\triangle ABC</math>, let <math>D, E</math> and <math>F</math> be points on <math>BC, AC</math> and <math>AB</math> respectively. The | + | For any triangle <math>\triangle ABC</math>, let <math>D, E</math> and <math>F</math> be points on <math>BC, AC</math> and <math>AB</math> respectively. The Generalized Wooga Looga Theorem (Gwoologth) or the Devil's Triangle Theorem states that if <math>\frac{BD}{CD}=r, \frac{CE}{AE}=s</math> and <math>\frac{AF}{BF}=t</math>, then <math>\frac{[DEF]}{[ABC]}=1-\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}=\frac{rst+1}{(r+1)(s+1)(t+1)}</math>. |
(*Simplification found by @Gogobao) | (*Simplification found by @Gogobao) | ||
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=Other Remarks= | =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: | 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 | + | https://webcache.googleusercontent.com/search?q=cache:Qoyk2gGO6x8J:https://artofproblemsolving.com/wiki/index.php/Wooga_Looga_Theorem+&cd=1&hl=en&ct=clnk&gl=us&client=safari |
+ | |||
Essentially, Wooga Looga is a special case of this, specifically when <math>r=s=t</math>. | Essentially, Wooga Looga is a special case of this, specifically when <math>r=s=t</math>. | ||
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trivial by <math>\frac{1}{2}ab\sin(C)</math> but ok ~ bissue | trivial by <math>\frac{1}{2}ab\sin(C)</math> but ok ~ bissue | ||
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+ | "Very nice theorem" - [[User:RedFireTruck|<font color="#FF0000">RedFireTruck</font>]] ([[User talk:RedFireTruck|<font color="#FF0000">talk</font>]]) 12:12, 1 February 2021 (EST) | ||
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+ | “I in the o’l days I used this theorem all the when time trying to tame my mammoth my cave buddy told me to ooga booga” - peelybonehead 9,000 B.C. | ||
+ | |||
+ | who else got redirected here from 2004 AMC 10B Problem 18 smh | ||
+ | |||
+ | I love the Wooga Looga Theorem! ~ Math-lover1 |
Latest revision as of 18:05, 23 March 2024
Contents
Definition
Generalized Wooga Looga Theorem (The Devil's Triangle)
For any triangle , let
and
be points on
and
respectively. The Generalized Wooga Looga Theorem (Gwoologth) 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 iswhere
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://webcache.googleusercontent.com/search?q=cache:Qoyk2gGO6x8J:https://artofproblemsolving.com/wiki/index.php/Wooga_Looga_Theorem+&cd=1&hl=en&ct=clnk&gl=us&client=safari
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)
“I in the o’l days I used this theorem all the when time trying to tame my mammoth my cave buddy told me to ooga booga” - peelybonehead 9,000 B.C.
who else got redirected here from 2004 AMC 10B Problem 18 smh
I love the Wooga Looga Theorem! ~ Math-lover1