**Problem**

In the *Zhoubi Suanjing*, the Gougu diagram as shown below demonstrates the Pythagorean theorem.

Use this diagram to prove the Pythagorean theorem in two ways.

**Solution**

The Gougu diagram provides two proofs. Let the length of the bold square be , the base of the right triangle is and the height of the right triangle .

**Proof 1**

The first proof uses the smallest square in the center and four right triangles that are contained in the bold square. The side length of the smallest square is . The area of the smallest square and the four triangles is equal to the area of the bold square.

**Proof 2**

The second proof uses the bolder square and the four right triangles outside the bold square. The area of the bold square and the four triangles is equal to the area of the largest square.