Equation (a(b-c)x^2+b(c-a)x+c(a-b)=0) has equal roots.
Discriminant = 0: [ [b(c-a)]^2-4\cdot a(b-c)\cdot c(a-b)=0 ]
Expanding and simplifying yields (2/b=1/a+1/c), i.e. (a,b,c) in H.P.
Proved.
Equation (a(b-c)x^2+b(c-a)x+c(a-b)=0) has equal roots.
Discriminant = 0: [ [b(c-a)]^2-4\cdot a(b-c)\cdot c(a-b)=0 ]
Expanding and simplifying yields (2/b=1/a+1/c), i.e. (a,b,c) in H.P.
Proved.