Pythagorean Theorem Proof (Thabit ibn Qurra)
Date: 03/28/2002 at 20:39:21 From: Natalie Bramlett Subject: Proving Pythag. th from cut-the-knot.com problem #24 Dr.Math, I am working on a proof from cut-the-knot.com, #24. I don't understand exactly what it is saying that I have to prove. I tried making the base triangle and 3,4,5 triangle and then going from there, but I don't know how they are forming the other triangles. If you could help me in any way I would greatly appreciate it. Thanks, Natalie
Date: 03/28/2002 at 23:26:24 From: Doctor Peterson Subject: Re: Proving Pythag. th from cut-the-knot.com problem #24 Hi, Natalie. I presume you are referring to http://www.cut-the-knot.org/pythagoras/ [Swetz] ascribes this proof to abu' l'Hasan Thabit ibn Qurra Marwan al'Harrani (826-901). It's the second of the proofs given by Thabit ibn Qurra. The first one is essentially the #2 above. The proof resembles part 3 from proof #12. ABC = FLC = FMC = BED = AGH = FGE. On the one hand, the area of the shape ABDFH equals AC^2 + BC^2 + area(ABC + FMC + FLC). On the other hand, area(ABDFH) = AB^2 + area(BED + FGE + AGH). F+ /| \ L / | + / | / \ E G+---+-------/-----------+ /| | / | \ D / | | / | + / | | / | / / | | / | / / | | / | / M+ | | / | / / \| |/ | / / | + C | / / | / \ | / + | / \ | / H \ |/ \ |/ +-----------------------+ A B I'm not sure what you mean about a 3-4-5 triangle; do you just mean that you used a 3-4-5 right triangle to draw the figure as an example? I believe the construction goes something like this: Draw right triangle ABC. Extend sides AC and BC beyond C by lengths MC=AC and CL = CB, forming squares ACMH and BCLD. Lines HM and DL will meet at F. Erect perpendiculars AG and BE to AB, with G and E on HF and DF respectively; you can show that this is a square. Now you can list various segments congruent to AB, BC, and AC (because they are part of of various parallelograms), and use those to prove the series of congruent triangles listed. (The vertices of these triangles are listed out of order; you will have to correct the naming.) The rest is just to dissect the whole figure two different ways. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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