Question

If a, b and c are sides of a triangle
Then prove that 1≤ [(a^2+b^2+c^2)/(ab + bc+ca)]

Answer 1

so, we want to show that

ab+bc+ca <= (a^2+b^2+c^2)

since all squares are positive,

0 <= (a-b)^2 + (a-c)^2 + (b-c)^2
0 <= a^2-2ab+b^2 + a^2-2ac+c^2 + b^2-2bc+c^2
0 <= 2(a^2+b^2+c^2) - 2(ab+ac+bc)
ab+ac+bc <= a^2+b^2+c^2

This is in fact true for any three real numbers, not just sides of a triangle