1. We are given the task to demonstrate the Pythagorean theorem: for a right triangle with legs $a$ and $b$, and hypotenuse $c$, prove that $$a^2 + b^2 = c^2.$$ 2. Consider a right triangle with sides $a$, $b$, and hypotenuse $c$. Construct a square with side length $a+b$. 3. Inside this square, arrange four copies of the triangle so that their hypotenuses form a smaller tilted square in the center with side length $c$. 4. The area of the large square is $$(a+b)^2 = a^2 + 2ab + b^2.$$ 5. The area can also be expressed as the sum of the areas of the four triangles and the smaller square: $$4 \cdot \frac{1}{2}ab + c^2 = 2ab + c^2.$$ 6. Equate the two expressions for the area: $$a^2 + 2ab + b^2 = 2ab + c^2.$$ 7. Subtract $2ab$ from both sides: $$a^2 + b^2 = c^2.$$ 8. This completes the proof of the Pythagorean theorem, showing that the sum of the squares of the legs equals the square of the hypotenuse.