According to the prompt, multiply both sides of A+B+C =ab+ac+bc by 2, and it will be converted into A 2a+ A 2B+ B 2C -2ab-2ac-2bc=0.

Formula (Party A+Party B -2AB)+ (Party B+Party C -2BC)+ (Party C+Party A -2ca)=0.

Then (a-b) square +(b-c) square +(c-a) square =0.

So a=b=c

So the triangle is an equilateral triangle.