Prove the condition that a triangle should be equilateral is that:
Here is another solution using the complex numbers:
The "corner" is actually the "corner" turned through with an angel of in the positive(or negative) direction!
Now, let us observe the complex number: .
This complex number is a solution of the equation: .
So:
By turning with positive direction we can describe algebraically:
Now the negative direction:
Hence again, algebraically:
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Now we simplify
or:
[1]
And same simplification on :
[2]
Now we multiply 1 and 2: