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February 22nd, 2013, 06:04 PM  #1 
Senior Member Joined: Apr 2012 Posts: 799 Thanks: 1  Area of triangle ABC
[attachment=0:3g7xy306]Area of triangle ABC.JPG[/attachment:3g7xy306]

February 22nd, 2013, 06:53 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1123 Math Focus: Elementary mathematics and beyond  Re: Area of triangle ABC 
February 22nd, 2013, 09:15 PM  #3 
Senior Member Joined: Apr 2012 Posts: 799 Thanks: 1  Re: Area of triangle ABC
greg1313: great ,you got it 
February 22nd, 2013, 10:47 PM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Area of triangle ABC
Letting the lowers case letters correspond to the sides opposite the given vertices, the law of cosines gives: By Pythagoras, we find: and so substituting into the first equation, we arrive at the factored quartic: Since as then: and the area is: 
February 23rd, 2013, 12:05 AM  #5 
Senior Member Joined: Apr 2012 Posts: 799 Thanks: 1  Re: Area of triangle ABC
now try to prove it with no trigonometry using geometry only 
February 23rd, 2013, 08:30 AM  #6 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1123 Math Focus: Elementary mathematics and beyond  Re: Area of triangle ABC
Let a, b and c be the sides opposite A, B and C. Let h be AH. Construct a circumcircle around ABC. By the Inscribed Angle theorem and some calculations, the radius of the circumcircle is 5/?2, so its diameter is 10/?2. Given that the diameter of the circumcircle is abc/(2T), where T is the area of the triangle, we have 10/?2 = 5bc/(2T). Since T = 5h/2, we have 10/?2 = bc/h. 50h² = b²c². b²  9 = c²  4, 50h² = 50(b²  9) = b²(b²  5) ? b = ?10, ?45. Only ?45 fits ABC, so T = 15. 
February 24th, 2013, 09:08 PM  #7 
Senior Member Joined: Apr 2012 Posts: 799 Thanks: 1  Re: Area of triangle ABC
The following graph may give you a hint : h=2r=6 the area is : [attachment=0:2nsrutdq]Area of ABC.JPG[/attachment:2nsrutdq] 

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