Prove the condition that a triangle should be equilateral is that:

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- Jan 5th 2011, 11:02 AMAlso sprach ZarathustraEquilateral Triangles.
Prove the condition that a triangle should be equilateral is that:

- Jan 5th 2011, 11:10 AMsnowtea
- Jan 5th 2011, 11:45 AMArchie Meade
- Jan 5th 2011, 04:12 PMSoroban
Hello, Also sprach Zarathustra!

Quote:

. .

We assume that are lengths of the sides (positive real numbers).

We have a quadratic equation:

. .

Apply the Quadratic Formula:

. .

This simplifies to:

. . .[1]

Since is a real number,

Then [1] becomes: .

- Jan 6th 2011, 05:48 AMAlso sprach Zarathustra

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:

================================================== =========

Now we simplify

or:

[1]

And same simplification on :

[2]

Now we multiply 1 and 2: