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: .
By turning with positive direction we can describe algebraically:
Now the negative direction:
Hence again, algebraically:
Now we simplify
And same simplification on :
Now we multiply 1 and 2: