The second discriminant of a univariate polynomial

We define the second discriminant D₂ of a univariate polynomial f of degree greater than 2 as the product of the linear forms 2r_k − r_i − r_j for all triples of roots r_i, r_k, r_j of f with i < j and j ≠ k, k ≠ i. D₂ vanishes if and only if f has at least one root which is equal to the average of two other roots. We show that D₂ can be expressed as the resultant of f and a determinant formed with the derivatives of f, establishing a new relation between the roots and the coefficients of f. We prove several notable properties and present an application of D₂.