Question

If `a, b and c` are the sides of the triangle then `(a^2 + b^2 +c^2)/(ab+bc+ca)` lies between?

Answer 1

`1 le (a^2+b^2+c^2)/(a b+b c+a c) lt 2`

Explanation:

Considering the area extrema for a triangle, the area is a maximum when the three sides are equal `a=b=c` and the area is a minimum when one side is equal to the sum of the other two `a = b = c/2+epsilon`, `epsilon > 0` and arbitrarily small, we have

(maximum area) `1 le (a^2+b^2+c^2)/(a b+b c+a c) < 6/5` (minimum area)

Now considering another case of null area in which one of the sides is arbitrarily small, `a = b > 0` and `c = epsilon` arbitrarily small, we can extend the range to

`1 le (a^2+b^2+c^2)/(a b+b c+a c) < 2`