Pythagorean Theorem

The document presents a proof attempt for the Pythagorean Theorem, starting from the assumption that a right triangle with legs a and b and hypotenuse c might violate the relation 𝑐 2 = 𝑎 2 + 𝑏 2 . The author explores two cases: Assuming 𝑐 2 > 𝑎 2 + 𝑏 2 The document shows that this leads to contradictions with the triangle inequality 𝑐 < 𝑎 + 𝑏. Several algebraic decompositions of the excess term 𝑟 are tested, each producing a contradiction such as 𝑐 = 𝑎 + 𝑏, which is impossible for a non-degenerate triangle. Assuming 𝑐 2 < 𝑎 2 + 𝑏 2 Similarly, this assumption leads to contradictions with the fact that the hypotenuse must be the longest side. Decompositions of the deficit term again produce impossible results such as 𝑐 = 𝑎 − 𝑏 or 𝑐 < 𝑎. Because both alternatives lead to contradictions, the document concludes that the only consistent possibility is: “𝑐 2 = 𝑎 2 + 𝑏 2” (from the document: “: c2 = a2 + b2 ... by A & B”) Thus, the proof uses proof by contradiction to establish the Pythagorean Theorem