  We prove that the probabilistic definition of the most general boundary
three-point and bulk-boundary structure constants in Liouville conformal field
theory (LCFT) agree respectively with the formula proposed by Ponsot-Techsner
(2002) and by Hosomichi (2001). These formulas also respectively 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
proof relies on the boundary Belavin-Polyakov-Zamolodchikov differential
equation recently proved by the first named author, and inputs from the
coupling theory of Liouville quantum gravity (LQG) and Schramm Loewner
evolution. Our results supply all the structure constants needed to perform the
conformal bootstrap for boundary LCFT. They also yield exact descriptions for
the joint law of the area and boundary lengths of basic LQG surfaces, including
quantum triangles and two-pointed quantum disks.
