Difference between revisions of "The Devil's Triangle"
Redfiretruck (talk | contribs) (→Proof 2) |
(→Proof 1) |
||
Line 23: | Line 23: | ||
Finally, we have <math>\frac{[DEF]}{[ABC]}=\boxed{1-\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}}</math>. | Finally, we have <math>\frac{[DEF]}{[ABC]}=\boxed{1-\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}}</math>. | ||
+ | (You can simplify it to <math>\frac{rst+1}{(r+1)(s+1)(t+1)}</math> ~ Gogobao) | ||
+ | ~@CoolJupiter | ||
− | |||
=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: |
Revision as of 10:57, 6 November 2020
Contents
[hide]Definition
For any triangle , let and be points on and respectively. Devil's Triangle Theorem states that if and , then .
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