Thread: Prove that triangle is equilateral

1. Prove that triangle is equilateral

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

Angles: $\displaystyle AFC=AEB=BDC= 120^o$

Thank you.

2. Re: Prove that triangle is equilateral

Hey feferon11.

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

3. 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.

4. 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.
Originally Posted by feferon11
Over the sides of an arbitrary triangle ABC from the outside are constructed isosceles triangles with an angle of $\displaystyle 120^o$ at the top.
Prove that the three vertices of this triangles form an equilateral triangle (triangle DFE on picture).

Angles: $\displaystyle AFC=AEB=BDC= 120^o$

Thank you.