$(I,r),$ $(I_a,r_a),$ $(I_b,r_b),$ $(I_c,r_c)$ denote the incircle and three excircles against $A,B,C.$ Let $(U_a,\rho_a),$ $(U_b,\rho_b),$ $(U_b,\rho_c)$ be the mixtilinear incircles against $A,B,C$ and $(V_a, \varrho_a),$ $(V_b,\varrho_b),$ $(V_c,\varrho_c)$ their corresponding mixtilinear excircles. $(U_a,\rho_a)$ and $(V_a, \varrho_a)$ are tangent to rays $AB,AC$ through $M,N$ and $X,Y$ respectively. It's well-known that lines $MN$ and $XY$ pass through the incenter $I$ and A-excenter $I_a,$ respectively. Thus, from the similar kites $AMU_aN \sim AXV_aY,$ it follows that $\alpha=\frac{\rho_a}{\varrho_a}=\frac{AI}{AI_a}=\frac{r}{r_a}$ Similarly, we have the ratios: $ \beta= \frac{\rho_b}{\varrho_b}=\frac{r}{r_b} \ , \ \gamma=\frac{\rho_c}{\varrho_c}=\frac{r}{r_c}$ $\Longrightarrow \alpha+\beta+\gamma=\frac{\rho_a}{\varrho_a}+\frac{\rho_b}{\varrho_b}+\frac{\rho_c}{\varrho_c}=r \left( \frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c} \right)=r \cdot \frac{1}{r}=1.$