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0|0|0|||||Dijkstra%27s proof of the Pythagorean Theorem|stevens||07:03:09|10/12/2009|Hello%21%0D%0A%0D%0AI enjoy this nice site very much.%0D%0A%0D%0AAfter reading about Dijkstra and the Pythagorean Theorem %0D%0A%28 http%3A%2F%2Fwww.cut-the-knot.org%2Fpythagoras%2FDijkstra.shtml %29%0D%0AI realised that the problem with his formula %0D%0Asgn%28%5Calpha%2B%5Cbeta-%5Cgamma%29%3Dsgn%28a%5E2%2Bb%5E2-c%5E2%29%0D%0Ais that it involves negative quantities. One should not construct the difference of angles%2C but their sum. For the picture this means placing the two smaller triangles on the outside. In the right-angled case one gets the figure of Proof %2341. Computing the  side lengths as in that Proof%2C one gets a simple proof of Dijkstra%27s formula. %0D%0AThe formula is a qualitative version of the more precise quantitative formula %0D%0Aa%5E2%2Bb%5E2-c%5E2%3D 2ab%5Csin 1%2F2%28%5Calpha%2B%5Cbeta-%5Cgamma%29%0D%0Awhich is basically the cosine rule. One can use the same picture to prove the rule.%0D%0A%0D%0AI have written a small text%2C with pictures%2C detailing the above. It is on my home page%0D%0Ahttp%3A%2F%2Fwww.math.chalmers.se%2F%7Estevens%2Fdijkstra.pdf%0D%0A%0D%0ABest wishes%2C%0D%0AJan Stevens%0D%0AMatematik%2C G%F6teborgs universitet
1|1|0|||||RE%3A Dijkstra%27s proof of the Pythagorean Theorem|alexb||09:56:49|10/12/2009|Jan%2C thank you for writing. Your diagram that extends proof %2341 is much more transparent than the one Dijkstra%27s proof.%0D%0A%0D%0ANow%2C could we proceed from here with first proving a%0D%0A%0D%0A%5Bh3%5DLemma%5B%2Fh3%5D%0D%0A%0D%0AIn an isosceles trapezium %28trapezoid here%2C in the US%29%2C that base is bigger which is incident to the smaller angles%2C and vice versa.%0D%0A%0D%0ANow%2C since the bases are parallel%2C the sum of the interior angles incident to one sideline is 180%26deg%3B%2C meaning that the sum of the pair of the base angles adjacent to the smaller base is more than that. By the Fifth postulate%2C the side lines meet beyond the base with the larger pair of angles. This proves the lemma because it produces a pair of similar triangles of which one is inside the other so that the sides of the interior triangle are shorter than the corresponding sides of the exterior triangle and so are the bases.%0D%0A%0D%0A%0D%0A
2|2|1|||||RE%3A Dijkstra%27s proof of the Pythagorean Theorem|stevens||13:57:57|10/14/2009|Your remark makes it clear that I should have written down the punch line. I considered the Lemma you stated as obvious%2C and considered my proof sketch complete after the picture.%0D%0A%0D%0ATo prove the Lemma I would%2C in the trapezoid ABCD with AB longer than DC%2C draw a line CE parallel to DA %28with E on AB%29. Then %5Ctriangle ECB is isosceles%2C so the angle B is acute. If conversely B is acute%2C the line CE will lie inside the figure%2C as the angle C is larger than its complementary angle%2C which is congruent to %5Cangle DCE. Therefore AB is longer than DC.
3|3|2|||||RE%3A Dijkstra%27s proof of the Pythagorean Theorem|alexb||14:34:27|10/14/2009|Yes%2C of course it is obvious. I have misconstrued your formula as a means of proving Dijkstra%27s identity. It is certainly a curiosity in its own right.%0D%0A%0D%0AI made it all into a page%0D%0A%0D%0Ahttp%3A%2F%2Fwww.cut-the-knot.org%2Fpythagoras%2FStevens.shtml%0D%0A%0D%0AI think your proof deserves a note in%2C say%2C Math Magazine or The Math Gazette.