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November 18th, 2012, 10:53 PM   #1
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area of square DEFG

[attachment=0:2dbl25o7]area of a square DEFG.JPG[/attachment:2dbl25o7]
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November 19th, 2012, 12:50 AM   #2
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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.
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November 19th, 2012, 09:16 AM   #3
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Re: area of square DEFG

One for you, Albert!
Using your diagram:
ADG = 864
BED = 1350
CGF = 2400

All sides = integers.

Sides of ABC = ?
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November 19th, 2012, 11:44 PM   #4
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Re: area of square DEFG

[attachment=0:3mojyc2q]area of square DEFG.JPG[/attachment:3mojyc2q]
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November 20th, 2012, 01:50 AM   #5
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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.
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November 20th, 2012, 04:36 AM   #6
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Re: area of square DEFG

Thanks : Hoempa
the error has been corrected
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November 20th, 2012, 05:17 AM   #7
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Re: area of square DEFG

ADG: 36-48-60 ; 864
BDE: 45-60-75 ; 1350
CFG: 60-80-100 ; 2400
DEFG: 60 by 60 ; 3600
====================
ABC: 111-148-185 ; 8214
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November 20th, 2012, 01:03 PM   #8
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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: 45-60-75.
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!
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November 20th, 2012, 05:27 PM   #9
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Re: area of square DEFG

Denis :
Many thanks
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