  We define a three-parameter family of random surfaces in Liouville quantum
gravity (LQG) which can be viewed as the quantum version of triangles. These
quantum triangles are natural in two senses. First, by our definition they
produce the boundary three-point correlation functions of Liouville conformal
field theory on the disk. Second, it turns out that the laws of the triangles
bounded by flow lines in imaginary geometry coupled with LQG are given by these
quantum triangles. In this paper we demonstrate the second point for boundary
flow lines on a quantum disk. Our method has the potential to prove general
conformal welding results with quantum triangles glued in an arbitrary way.
Quantum triangles play a basic role in understanding the integrability of SLE
and LQG via conformal welding. In this paper, we deduce integrability results
for chordal SLE with three force points, using the conformal welding of a
quantum triangle and a two-pointed quantum disk. In a subsequent work we will
explore their applications to the mating-of-trees framework of LQG, including
the exact evaluation of the expected proportion of inversions in skew Brownian
permutons.
