can you generate and explain a proof of the pythagorean theorem
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Thinking Summary
**Generating a Pythagorean theorem proof** The user wants me to generate and explain a proof of the Pythagorean theorem. I need to create a clear proof, possibly using methods like rearrangement or similar triangles. It’s important to provide a step-by-step explanation to ensure the user understands. I’ll make sure to format any mathematical parts using KaTeX to enhance clarity. Let’s craft a well-structured response that answers their question thoroughly!
Below is a classic, purely geometric proof of the Pythagorean Theorem (often called the “re-arrangement” or “Bhāskara” proof), followed by an explanation of each step.
Statement of the theorem
For any right triangle with legs of lengths and and hypotenuse of length ,
Construction
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Draw a right triangle with right angle at .
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Make four identical copies of .
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Arrange the four copies inside a large square of side length in the following way:
• Place one triangle in each corner so that the right angles touch the corners of the large square.
• Orient each triangle the same way (legs along the square’s edges, hypotenuse facing inward).After the placement, the four hypotenuses form a smaller, tilted square in the center.
Areas
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Area of the big square
Each side is , so
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Area accounted for by the four triangles
Each triangle has area , so four of them have total area
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Area of the inner (tilted) square
What remains after removing the triangles from the big square is the inner square. Its area is therefore
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But the inner square’s side length is
Because each side of the inner square is exactly the hypotenuse of one triangle,
Equating the two expressions for the inner square
Set the two formulas obtained for the inner square equal:
Expand the right–hand side:
Hence,
This completes the proof.
Why the argument works
• Each “piece” (big square, four triangles, inner square) is counted exactly once; no overlaps or gaps exist because the triangles’ right angles exactly fill the corners.
• The hypotenuse of every triangle is straight, so the four together form a true square centrally.
• Equating the two descriptions of the same inner region’s area forces the algebraic relationship .
Historical note
A version of this rearrangement proof appears in Chinese texts (the Zhoubi Suanjing) and in Bhāskara II’s 12-century work in India. Bhāskara famously wrote a single word next to the diagram: “Behold!” — inviting the reader to see the result almost at a glance.