Difference between revisions of "The Devil's Triangle"
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− | 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. Devil's Triangle Theorem, also known has Routh 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)}</math> | + | 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. Devil's Triangle Theorem, also known has Routh 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)}</math>, or <math>\frac{rst+1}{(r+1)(s+1)(t+1)</math>(Shoutout to @Gogobao for pointing this out) |
=Proofs= | =Proofs= |
Revision as of 15:32, 6 November 2020
Contents
[hide]Definition
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 $\frac{rst+1}{(r+1)(s+1)(t+1)$ (Error compiling LaTeX. Unknown error_msg)(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 . (You can simplify it to ~ Gogobao) ~@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
The Ooga Booga Tribe would be proud of you. Amazing theorem - RedFireTruck
This is Routh's theorem isn't it~ Ilovepizza2020