The Devil's Triangle
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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 .
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 .