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February 22nd, 2013, 07:04 PM   #1
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Area of triangle ABC

[attachment=0:3g7xy306]Area of triangle ABC.JPG[/attachment:3g7xy306]
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February 22nd, 2013, 07:53 PM   #2
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Re: Area of triangle ABC

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February 22nd, 2013, 10:15 PM   #3
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Re: Area of triangle ABC

greg1313: great ,you got it
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February 22nd, 2013, 11:47 PM   #4
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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:

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February 23rd, 2013, 01:05 AM   #5
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Re: Area of triangle ABC

now try to prove it with no trigonometry

using geometry only
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February 23rd, 2013, 09:30 AM   #6
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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. 50h = bc. b - 9 = c - 4, 50h = 50(b - 9) = b(b - 5) ? b = ?10, ?45. Only ?45 fits ABC, so T = 15.
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February 24th, 2013, 10:08 PM   #7
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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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