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0|0|0|||||Maximum Area Property of Equilateral Triangles|Bui Quang Tuan||23:53:40|08/11/2010|Dear Alex%2C%0D%0A%0D%0AThe message is related Maximum Area Property of Equilateral Triangles%3A%0D%0Ahttp%3A%2F%2Fwww.cut-the-knot.org%2Ftriangle%2FInscribedEquilateral.shtml%0D%0A%0D%0AI think after proofing that isosceles triangle has area bigger than any triangle with common base%2C we can have another proof as following by infinite continue of changing apex of isosceles triangles.%0D%0A%0D%0AWe start with any triangle ABC inscribed in given circumcircle and infinite continue of making the triangle isosceles with apex%3A A%2C B%2C C%2C A%2C B%2C C...%0D%0A%0D%0AAt least one of three angles A%2C B%2C C must be less or equal 60 %28degree%29 so we can suppose A %3C%3D 60.%0D%0A%0D%0AIf we make the triangle isosceles at one angle %28vertex X%29 so these angle X is not changed%2C only two other changed and equal %28180 - X%29%2F2%0D%0A%0D%0AThe angles we can get after each time as following%3A%0D%0AA%2C B%2C C%0D%0AA%2C %28180-A%29%2F2%2C %28180-A%29%2F2%0D%0A%28180-%28180-A%29%2F2%29%2F2%2C %28180-A%29%2F2%2C %28180-%28180-A%29%2F2%29%2F2%0D%0A...%0D%0A%0D%0ASo%2C we get a sequence a%28n%29 of angles%3A%0D%0AA%2C %28180-A%29%2F2%2C %28180-%28180-A%29%2F2%29%2F2%2C...%0D%0A%0D%0Ahere%3A a%280%29%3DA%2C a%28n%2B1%29 %3D %28180-a%28n%29%29%2F2 with n%3D1%2C2%2C3...%0D%0A%0D%0AIf 0%3C%3DX%3C%3D60 then X%3C%3D%28180-X%29%2F2 and %28180-X%29%2F2%3E%3D60%0D%0AIf X%3E%3D60 then X%3E%3D%28180-X%29%2F2 and %28180-X%29%2F2%3C%3D60%0D%0AIf 0%3C%3DX%3C%3D60 then %28180-%28180-X%29%2F2%29%2F2%3E%3DX and %28180-%28180-X%29%2F2%29%2F2 %3C%3D60%0D%0AIf X%3E%3D60 then %28180-%28180-X%29%2F2%29%2F2%3C%3DX and %28180-%28180-X%29%2F2%29%2F2 %3E%3D60%0D%0A%0D%0ASince we choose A%3C%3D60 so a%28n%29 can be devided into two sub monotone sequences%3A%0D%0Aa%280%29%3C%3Da%282%29%3C%3Da%284%29%3C%3Da%286%29%3C%3D... %3C%3D60%0D%0Aa%281%29%3E%3Da%283%29%3E%3Da%285%29%3E%3Da%287%29%3E%3D... %3E%3D60%0D%0A%0D%0ATherefore these two subsequences have limits. Since both have common form f%3D%28180-f%29%2F2 so they have common limit 60. It means we can get%2C at the end%2C equilateral triangle and equilateral triangle has maximum area%0D%0A%0D%0ABest regards%2C%0D%0ABui Quang Tuan