Proof of the Pythagorean Theorem Statement
Reason
Consider a right-angled triangle with sides $a$, $b$, and hypotenuse $c$.
Given.
Construct a square with side length $a+b$.
Geometric construction.
Inside this square, arrange four copies of the triangle and a smaller square of side $c$.
Visual arrangement.
The area of the large square is $(a+b)^2$.
Area of a square formula.
The area of the four triangles is $4 \times \frac{1}{2}ab = 2ab$.
Area of a triangle formula.
The area of the inner square is $c^2$.
Area of a square formula.
The area of the large square is also the sum of the areas of the four triangles and the inner square:
$$ (a+b)^2 = 2ab + c^2 $$
Area decomposition principle.
Expanding the left side:
$$ a^2 + 2ab + b^2 = 2ab + c^2 $$
Algebraic expansion.
Subtract $2ab$ from both sides:
$$ a^2 + b^2 = c^2 $$
Algebraic manipulation.
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