Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle

Proof

 * Euclid-XIII-12.png

Let $ABC$ be a circle.

Let the equilateral triangle $ABC$ be inscribed within the circle $ABC$.

It is to be demonstrated that the square on one side of $\triangle ABC$ is $3$ times the square on the radius of the circle $ABC$.

Let $D$ be the center of the circle $ABC$.

Let $AD$ be joined and produced to $E$.

Let $BE$ be joined.

We have that $\triangle ABC$ is equilateral.

Therefore the arc $BEC$ is a third part of the circumference of the circle $ABC$.

Therefore the arc $BE$ is a sixth part of the circumference of the circle $ABC$.

Therefore $BE$ is the side of a regular hexagon inscribed within the circle $ABC$.

Therefore from :
 * $BE = DE$

We have that:
 * $AE = 2 \cdot DE$

So:
 * $AE^2 = 4 \cdot ED^2 = 4 \cdot BE^2$

But from :

and:

it follows that:
 * $AE^2 = AB^2 + BE^2 = 4 \cdot BE^2$

Therefore:
 * $AB^2 = 3 \cdot BE^2$

But:
 * $BE = DE$

Therefore:
 * $AB^2 = 3 \cdot DE^2$

Hence the result.