Because the sum of two sides is greater than the third side, it is impossible for a+b = c.

So it can only be a=b, which is an isosceles triangle.

Only the conclusion that a=b can be reached, but there must be other conditions (such as an internal angle of 60 degrees) to get an equilateral triangle.

Your hypothesis is correct, but this is only a special case. You can't push back, you must push at once.