Because =BB'=CC', AC+AA'=BC+CC'=AB+BB' means A'C=BC'=AB', and Angle A'CC'= Angle A'AB'= Angle B'BC'

So triangles a' cc', a' b' a and b' BC' are congruent.

so∠ca ' c '+∠b ' a ' a =∠a ' b ' a+∠bb ' c ' =∠a ' c ' c+∠b ' c ' c '

So the three angles of the triangle add up to 180.

So ∠B'A'C'=∠A'C'B'=∠A'B'C'

So ... △A'B'C' is an equilateral triangle.