Beautiful Geometry 1

Consider the inscribed trapezoid $ABCD$. Let $M$ be the intersection of its diagonals and let $O$ be the center of the circle.

Since $ABCD$ is isosceles, we have $∠CAB = ∠ABD = 45$, and then $∠AMD = 90$. Also, $∠AOD = 2∠ABD = 90$. Now, applying the Pythagorean Theorem to $△AMD$ and $△AOD$, we get:

$S = 2S(ABM) + 2S(MCD) = AM^2 + MD^2 = AD^2 = AO^2 + OD^2 = 2.$