Difference of Two Squares/Geometric Proof 2

Proof

 * Difference-of-Two-Squares.png

Let $\Box ABCD$ be a square of side length $x$.

Let $\Box DEFG$ be a square of side length $y$ where $y < x$

Let $EF$ be produced to $H$.

The area of $\Box ABCD$ is seen to be equal to the sum of:
 * the area of the rectangle $AEHB$
 * the area of the rectangle $FGCH$
 * the area of the square $DEFG$

From Area of Square:
 * the area of $\Box ABCD$ is equal to $x^2$
 * the area of $\Box DEFG$ is equal to $y^2$

From Area of Rectangle:
 * the area of $AEHB$ is equal to $x \paren {x - y}$
 * the area of $FGCH$ is equal to $y \paren {x - y}$

Hence: