If a, b, c are the lengths of the sides of a triangle ABC, then ab+bc+ca/a^2+b^2+c^2 lies in the interval

The upper limit is when the triangle is equilateral. The value of the expression in this case is 1.

The lower limit is for the triangle (imaginary) for which the angles are 90 degrees, 90 degrees and 0 degrees. a = 1, b = 1 and c = 0 in this case. The value of the expression in this case is 1/2. But this value is never attained. So, the range of the expression is (0.5, 1]