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November 18th, 2012, 10:53 PM  #1 
Senior Member Joined: Apr 2012 Posts: 799 Thanks: 1  area of square DEFG
[attachment=0:2dbl25o7]area of a square DEFG.JPG[/attachment:2dbl25o7]

November 19th, 2012, 12:50 AM  #2 
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  Re: area of square DEFG
Triangle ADG and ABC are similar. When triangles BED and CFG are joined on sides ED and FG then the "new" triangle is similar to ADG is four times larger in area. Hence BC  FE = 2*DG = 2*EF and so BC = 3*EF and the area of the square is 18  2  3  5 = 8. 
November 19th, 2012, 09:16 AM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,274 Thanks: 1023  Re: area of square DEFG
One for you, Albert! Using your diagram: ADG = 864 BED = 1350 CGF = 2400 All sides = integers. Sides of ABC = ? 
November 19th, 2012, 11:44 PM  #4 
Senior Member Joined: Apr 2012 Posts: 799 Thanks: 1  Re: area of square DEFG
[attachment=0:3mojyc2q]area of square DEFG.JPG[/attachment:3mojyc2q]

November 20th, 2012, 01:50 AM  #5 
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  Re: area of square DEFG
Thanks, [color=#00AA00]skipjack[/color]! Unfortunately I make these slips from time to time. Albert made some too, Triangle ADG isn't similar to Triangle DBF and 2/8 <> 4. 
November 20th, 2012, 04:36 AM  #6 
Senior Member Joined: Apr 2012 Posts: 799 Thanks: 1  Re: area of square DEFG
Thanks : Hoempa the error has been corrected 
November 20th, 2012, 05:17 AM  #7 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,274 Thanks: 1023  Re: area of square DEFG
ADG: 364860 ; 864 BDE: 456075 ; 1350 CFG: 6080100 ; 2400 DEFG: 60 by 60 ; 3600 ==================== ABC: 111148185 ; 8214 
November 20th, 2012, 01:03 PM  #8 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,274 Thanks: 1023  Re: area of square DEFG
By the way Albert, there really was no computer search involved: there is ONLY one pythagorean triangle with area 1350: 456075. From that, CF = 80; rest is a breeze! If you want to see a REALLY FULLY INTEGRALized one: AB = 169, AC = 195, BC = 182. The square is a 84 by 84. Height of ABC also integral: 156. AN (N on BC) is height line: crosses DG at M: triangles ADM, AGM, ABN and ACN are all integer sided! 
November 20th, 2012, 05:27 PM  #9 
Senior Member Joined: Apr 2012 Posts: 799 Thanks: 1  Re: area of square DEFG
Denis : Many thanks 

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