If `a^2(b+c),b^2(c+a),c^2(a+b),` are in A.P. show that either `a ,b ,c` are in A.P., or `a b+b c+c a=0.`

Explanation

Given:
`a2(b+c),b2(c+a),c2(a+b)` are in AP


=> difference between second and first term = difference between third and second term


`=>b^2(c+a)−a2(b+c)=c2(a+b)−b2(c+a)`


`=>b^2c+ab^2−a^2b−a^2c=ac^2+bc^2−b^2c−b^2a`


...