Difference between revisions of "The Devil's Triangle"
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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>. | ||
+ | |||
+ | ~@CoolJupiter | ||
=Other Remarks= | =Other Remarks= |
Revision as of 09:39, 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 .
Proof
Proof 1
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
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 .