Distinguishing between the twin $b$-flavored unitarity triangles on a circular arc

With the help of the generalized Wolfenstein parametrization of quark flavor mixing and CP violation, we calculate fine differences between the twin $b$-flavored unitarity triangles defined by $V^{}{ub} V^{}{ud} + V^{}{cb} V^{}{cd} + V^{}{tb} V^{}{td} = 0$and$V^{}{ud} V^{}{td} + V^{}{us} V^{}{ts} + V^{}{ub} V^{}{tb} = 0$ in the complex plane. We find that vertices of the rescaled versions of these two triangles, described respectively by $\bar{\rho} + {\rm i} \bar{\eta} = -\left(V^{}{ub} V^{}{ud}\right)/\left(V^{}{cb} V^{}{cd}\right)$and$\widetilde{\rho} + {\rm i} \widetilde{\eta} = -\left(V^{}{ub} V^{}{tb}\right)/\left(V^{}{us} V^{}{ts}\right)$, are located on a circular arc whose center and radius are given by $O=(0.5,0.5\cot \alpha )$ and $R=0.5\csc \alpha$ with $\alpha$ being their common inner angle. The small difference between $(\bar{\rho },\bar{\eta })$ and $(\tilde{\rho },\tilde{\eta })$ is characterized by $\widetilde{\rho} - \bar{\rho} \sim \widetilde{\eta} - \bar{\eta} \sim {\cal O}(\lambda^2)$with$\lambda \simeq 0.22$ being the Wolfenstein expansion parameter, and these two vertices are insensitive to the two-loop renormalization-group running effects up to the accuracy of ${\cal O}(\lambda^4)$. Some comments are also made on similar features of three pairs of the rescaled unitarity triangles of lepton flavor mixing and CP violation.