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Math Help - Prove that triangle is equilateral

  1. #1
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    Prove that triangle is equilateral

    Over the sides of an arbitrary triangle ABC from the outside are constructed isosceles triangles with an angle of 120^o at the top.
    Prove that the three vertices of this triangles form an equilateral triangle (triangle DFE on picture).


    Angles: AFC=AEB=BDC= 120^o

    Thank you.
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  2. #2
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    Re: Prove that triangle is equilateral

    Hey feferon11.

    Hint: Can you prove that all interior angles are 60 degrees?
    Thanks from feferon11
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  3. #3
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    Re: Prove that triangle is equilateral

    Hi,
    Your problem is a rather famous theorem, namely Napoleon's Theorem. There are close to a zillion proofs of this theorem. You can start at wikapedia's article to start examining the various proofs. I've attached a sketch and a brief indication why your problem is Napoleon's.

    Prove that triangle is equilateral-mhfnapoleon.png
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  4. #4
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    Re: Prove that triangle is equilateral

    Chiro has given a very good hint.
    angle FCD = 120 given. triangle FCD is an isosceles triangle thus angle CFD = angle CFD = 30 degree. similarly from triangle EAF we will get
    angle AEF = angle AFE = 30 degree
    We have angle AFC = 120
    I am sure now you can establish that angle EFD = 60 degree and similarly other angles.
    Quote Originally Posted by feferon11 View Post
    Over the sides of an arbitrary triangle ABC from the outside are constructed isosceles triangles with an angle of 120^o at the top.
    Prove that the three vertices of this triangles form an equilateral triangle (triangle DFE on picture).


    Angles: AFC=AEB=BDC= 120^o


    Thank you.
    Thanks from feferon11
    Follow Math Help Forum on Facebook and Google+

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