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According to historical documents, it is challenging to establish whether a proof of Th¯abit’s theorem exists based exclusively on equidecomposibility, as in the case of the Pythagorean and Pappus theorems. This article presents the corresponding proof. Key Words: Euclidean geometry, generalization of Pythagorean theorem, equide-
Many people had commented on the Pythagorean Theorem, but Thabit ibn Qurra (b. 836 in Turkey, d. 02.18.901 in Iraq) was one of the first to offer commentary on it and create a new proof for it.
the Pythagorean theorem is a special. case, one can write. 52 _|_ C2 = a2 _|_ 26c cos A. 2. Upon substituting for a, b, and c, respectively, in step 1, the result is: (AB)2 + (AC)2 = (BC)2 + 2(AB)(AC) cos A. 3. Since angle BAC = angle AC'B' = angle AB'C, then cos A = cos AB'C = cos AC'B'.
The proof of Thabit’s generalization is identical to one of the standard ways of proving the theorem of Pythagoras using similar triangles, and provides a very accessible generalisation with which to challenge learners.
By the Pythagorean Theorem (applied twice), the expression |AB |2 + |AC |2 = b 2 + c 2 is equal to 2h 2 + (p + y)2 + (q + y)2. Since the Pythagorean Theorem also yields the equation y 2 + h 2 = r 2, it follows that b 2 + c 2 = 2r 2 – 2y 2 + (p + y) 2 + (q + y) 2 = p 2 + q 2 + 2 p y + 2q y + 2 r 2. Now consider the other side of the equation ...
His work in mathematics also includes original treatises, with contributions in the many areas of geometry and number theory. His original contributions include proofs of the Pythagorean theorem, a proof of Menelaus's theorem, proofs of Euclid's fifth postulate, and work on composite ratios.
15 Δεκ 2009 · Thabit ibn Qurra (826–901) was one of history’s most original thinkers and displayed expertise in the most difficult disciplines of this time: geometry, number theory, and astronomy as well as ontology, physics, and metaphysics.
