equiangular triangle and law of cosine

Question

How can I prove that Equiangular triangles are equilateral in plane geometry using Law of cosine? How is it that equiangular triangles are equilateral is a direct consequence of law of cosine?

Answer 1

ápplying the law of cosines we obtain $$a^2=b^2+c^2-bc$$ $$b^2=a^2+c^2-ac$$ $$c^2=a^2+b^2-ab$$ adding these equations we get $$a^2+b^2+c^2-ab-bc-ac=0$$ and this is $$\frac{1}{2}\left((a-b)^2+(b-c)^2+(c-a)^2\right)=0$$ and this is what you want.