The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse $c$ is equal to the sum of the squares of the other two sides $a$ and $b$, or:

$c^2 = a^2 + b^2$

Proof using Geometry (Rearrangement Method):

1. Construct a Square with Side Length $a+b$:
   Create a large square where each side has length $a+b$. Within the square, place four identical right triangles with legs $a$ and $b$ and hypotenuse $c$.

2. Area of the Large Square:
   The total area of the large square is:
   $(a+b)^2$

3. Alternative Area Representation:
   The large square can also be expressed as the sum of the areas of the four triangles and the smaller square left in the middle (whose side is $c$):
   $\text{Area} = 4 \times \left(\frac{1}{2}ab\right) + c^2 = 2ab + c^2$

4. Equating the Two Area Representations:
   Since the area is the same either way, we equate the expressions:
   $(a+b)^2 = 2ab + c^2$

5. Simplify the Equation:
   Expand $(a+b)^2$:
   $a^2 + 2ab + b^2 = 2ab + c^2$ Cancel out $2ab$ from both sides:
   $a^2 + b^2 = c^2$

Thus, we have proved the Pythagorean Theorem.

---

Alternate Proof (Algebraic)

Using similar triangles, the theorem can also be proved by comparing ratios. If you'd like, I can elaborate on this or other proofs.

---

Do you want further explanation or additional proofs?

Related Questions:

1. Can the Pythagorean Theorem be extended to three dimensions?
2. How is the theorem used in coordinate geometry?
3. What is the converse of the Pythagorean Theorem?
4. Are there non-Euclidean versions of the theorem?
5. How is the Pythagorean Theorem used in trigonometry?

Tip:

Visual proofs, like the one above, are excellent for understanding geometric concepts more intuitively.