Square Inscribed in Circle is greater than Half Area of Circle

Theorem
A square inscribed in a circle has an area greater than half that of the circle.

Proof

 * InscribedCircumscribedSquare.png

Let $ABCD$ be a square inscribed in a circle.

Let $EFGH$ be a square circumscribed around the same circle.

We have that:
 * $ABCD$ is twice the area of the triangle $ADB$.
 * $EFGH$ is twice the area of the rectangle $EFBD$.

From Area of Rectangle, the area of $EFBD$ is $ED \cdot DB$.

From Area of Triangle in Terms of Side and Altitude, the area of $ADB$ is $\dfrac 1 2 \cdot ED \cdot DB$.

Thus:
 * $EFGH = 2 \cdot ABCD$

But the area of $EFGH$ is greater than the area of the circle around which it is circumscribed.

Therefore half of the area of $EFGH$ is greater than half of the area of the circle around which it is circumscribed.

Therefore the area of $ABCD$ is greater than half of the area of the circle within which it is inscribed.

Historical Note
This result is used in, specifically Book $\text{XII}$, in several places. However, it is never extracted and proved separately as a lemma; the result is merely stated without comment after the first time it is demonstrated.

On it has been determined as being an important enough result to be proved in its own right.