Question

If a, b, c are the sides of a triangle then the range of (ab + bc + ca)/(a² + b² + c²) is

Answer 1

For a triangle \(\frac a{sin A}=\frac{b}{sinB}=\frac{c}{sin C}\)

Therefore

a = \(\frac{c(sin A)}{sin C}\) and b = \(\frac{c(sin B)}{sin C}\) 

By substituting we get the above expression as

Now we know that for maximum value of triangle must be equilateral

Substituting sinA = sinB = sinC = \(\frac{\sqrt3}2\), we get \(\frac12[\frac93-1]\) 

 = \(\frac12\)[3 - 1]

 = 1