1. The problem asks why the equation $a^2 + b^2 = c^2$ holds true. 2. This equation is known as the Pythagorean theorem, which applies to right-angled triangles. 3. The theorem states that in a right triangle, the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the lengths of the other two sides ($a$ and $b$). 4. The formula is: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse, and $a$ and $b$ are the legs. 5. This relationship can be proven using geometric methods, such as rearranging squares on the sides of the triangle or using similarity of triangles. 6. Intuitively, it shows how the area of the square built on the hypotenuse equals the sum of the areas of the squares built on the other two sides. 7. This theorem is fundamental in geometry and helps calculate distances and lengths in various applications.