We have analyzed the singularities of a triangle loop integral in detail and derived a formula for an easy evaluation of the triangle singularity on the physical boundary. It is applied to the Λb→J/ψK−p process via Λ∗-charmonium-proton intermediate states. Although the evaluation of absolute rates is not possible, we identify the χc1 and the ψ(2S) as the relatively most relevant states among all possible charmonia up to the ψ(2S). The Λ(1890)χc1p loop is very special as its normal threshold and triangle singularities merge at about 4.45 GeV, generating a narrow and prominent peak in the amplitude in the case that the χc1p is in an S-wave. We also see that loops with the same charmonium and other Λ∗ hyperons produce less dramatic peaks from the threshold singularity alone. For the case of χc1p→J/ψp and quantum numbers 3/2− or 5/2+ one needs P- and D-waves, respectively, in the χc1p, which drastically reduce the strength of the contribution and smooth the threshold peak. In this case we conclude that the singularities cannot account for the observed narrow peak. In the case of 1/2+, 3/2+ quantum numbers, where χc1p→J/ψp can proceed in an S-wave, the Λ(1890)χc1p triangle diagram could play an important role, though can neither assert their strength without further input from experiments and lattice QCD calculations.