We show that the probabilistic definition of the boundary three-point and bulk-boundary structure constants in Liouville conformal field theory (LCFT) agree with the formulas proposed by Ponsot-Teschner (2002) and Hosomichi (2001) respectively. These formulas also describe the fusion kernel and modular kernel of the Virasoro conformal blocks, which are important functions in various contexts of mathematical physics. As an intermediate step, we obtain the formula for the boundary reflection coefficient of LCFT proposed by Fateev-Zamolodchikov-Zamolodchikov (2000). Our argument depends on the boundary Belavin-Polyakov-Zamolodchikov differential equation, and inputs from the coupling theory of Liouville quantum gravity and Schramm-Loewner evolution. Our results supply all the structure constants needed to perform the conformal bootstrap for boundary LCFT.