Triangle inequality Bures distance

Question

Given the Bures distance between two density matrices $D_B(\rho_1,\rho_2) = 2(1-\|\rho_{1}^{1/2}\rho_2^{1/2}\|)$ where $\|\cdot\|$ is the norm defined by $\|A\| = \mathrm{Tr}(\sqrt{A^\star A})$, I try to show that this obeys the triangle inequality. Let $\rho_1,\rho_2,\rho_3$ be density matrices. We need that $D_B(\rho_1,\rho_2)\leq D_B(\rho_1,\rho_3) + D_B(\rho_2,\rho_3)$. My attempt is to show that $\|\rho_1^{1/2} \rho_3^{1/2}\| + \|\rho_{2}^{1/2} \rho_3^{1/2}\| \leq 1+\|\rho_1^{1/2} \rho_2^{1/2}\|$. I know that $\|\rho_{i}^{1/2}\rho_{j}^{1/2}\| \leq 1$ for any $i,j$. However, this is not sufficient for the proof. I am clearly overlooking something. Could someone give me a hint?