  There are six leptonic unitarity triangles (LUTs) defined by six
orthogonality conditions of the three-family lepton flavor mixing matrix in the
complex plane. In the framework of the standard model or the minimal
supersymmetric standard model, the evolutions of sides and inner angles of the
six LUTs from a superhigh energy scale $\Lambda_{\rm H}^{}$ to the electroweak
scale $\Lambda_{\rm EW}^{}$ due to the renormalization-group equation (RGE)
running are derived in the integral form for both Dirac and Majorana neutrinos.
Furthermore, the LUTs as an intuitively geometrical language are applied to the
description of the RGE-induced $\mu$-$\tau$ reflection symmetry breaking
analytically and numerically.
