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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 at the top.

Prove that the three vertices of this triangles form an equilateral triangle (triangle DFE on picture).

Angles:

Thank you.

Re: Prove that triangle is equilateral

Hey feferon11.

Hint: Can you prove that all interior angles are 60 degrees?

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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.

Attachment 28614

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** Over the sides of an arbitrary triangle ABC from the outside are constructed isosceles triangles with an angle of

at the top.

Prove that the three vertices of this triangles form an equilateral triangle (triangle DFE on picture).

Angles:

Thank you.