An operator triangle inequality for the quadratic symmetric modulus

50 years after Thompson’s famous triangle inequality for the operator right modulus, we establish a triangle inequality for the quadratic symmetric modulus. We also discuss the corresponding equality cases as well as the infinite-dimensional setting. In addition, we obtain Clarkson–McCarthy type inequalities for the quadratic symmetric modulus. Moreover, we answer several questions raised by Bourin and Lee in [\emph{Bull. Lond. Math. Soc.} $\mathbf{\text{44}}$ (2012), no.~6, 1085–1102] and [\emph{Internat. J. Math.} $\mathbf{\text{31}}$ (2026), no.~6, 2650018].