The condition for the equation Ax^2 + Bx + C = 0 to have equal roots is that its discriminant must be zero, meaning B^2 - 4AC = 0. Substituting the given expressions yields (b^2c + bc^2 - a^2b - ab^2)^2 - 4(a^2b + ab^2 - a^2c - ac^2)(a^2c + ac^2 - b^2c - bc^2) = 0. Factoring the terms inside the formula gives (bc(b+c) - ab(a+b))^2 - 4(ab(a+b) - ac(a+c))(ac(a+c) - bc(b+c)) = 0. By applying the algebraic identity that (x-y)^2 - 4(y-z)(z-x) is equivalent to (x+z-2y)^2, we establish that a+b-2c = 0. The relation a+c = 2b means that a, b, and c form an arithmetic progression, which implies that their reciprocals 1/a, 1/b, and 1/c form a harmonic progression.